problem solving using mean

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

In order to access this I need to be confident with:

This topic is relevant for:

Here we will learn about the mean, including what the mean is and how to find the mean.

There are also mean worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is the mean?

The mean is a type of average calculated by finding the total of the values and dividing the total by the number of values.

The “number of values” is sometimes referred to as the “number of numbers”.

E.g. Work out the mean

The mean is 9.4

The mean (also known as the arithmetic mean) is a measure of central tendency because it describes a set of numbers by identifying a central position within the data.

How to calculate the mean

In order to calculate the mean:

Find the total of the values.

Divide the total by the number of values.

Write down the answer.

Mean, median, mode and range worksheet

Get your free mean maths worksheet of 20+ mean, median, mode and range questions and answers. Includes reasoning and applied questions.

Related lessons on mean, median, mode

Mean is part of our series of lessons to support revision on mean, median, mode . You may find it helpful to start with the main mean, median, mode lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Mean, median, mode
• Mode or modal

Mean maths examples

Example 1: finding the mean.

Calculate the mean value of this list of numbers:

Add up all the values in the list

2 Divide the total by the number of values.

There are 5 values in the data set. We need to divide the total by 5 .

3 Write down the answer.

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures.

Here the mean is an integer, so there is no need to round.

The mean is 8

Example 2: finding the mean

Calculate the mean value of this set of numbers.

Give your answer to 1 decimal place.

There are 6 values in the data set. We need to divide the total by 6 .

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures. Here we have been asked to round to 1 decimal place.

Example 3: finding the mean

Calculate the mean value of this set of data. Give your answer to 1 decimal place.

Add up all the values in the list 

There are 7 values in the data set.  We need to divide the total by 7 .

If your answer is a decimal, you may need to round your answer to a given number of decimal places or significant figures. 

Here we have been asked to round to 1 decimal place.

Example 4: finding the mean

Calculate the mean value of this list of numbers.

We have not been asked to round and the mean is a simple decimal.

The mean is 104.4

How to solve a problem involving the mean

In order to solve a problem involving the mean:

Use the mean and number of values to find the total.

Read the question carefully to work the next step.

Find the answer.

Problem solving involving mean examples

Example 5: problem solving.

The mean of 4 values is 10 .

Here are 3 of the values:

Find the 4 th value.

The mean of 4 values is 10 .  We can multiply these together to find the total of the 4 numbers.

We need to find the 4

th value. We can find the total of the first 3 values.

Then we can subtract this total from the total of the 4 values.

We need to find the difference of the totals.

The 4 th value is 13

Alternatively, you could use the equation for finding the mean. You could use x as the missing value.

Then rearrange and solve.

Example 6: problem solving

The mean of 5 values is 14 .

Here are 4 of the values:

Find the 5 th value:

The mean of 5 values is 14 . We can multiply these together to find the total of the 5 numbers.

We need to find the 5 th value. We can find the total of the first 4 values. 

Then we can subtract this total from the total of the 5 values.

The 5 th value is 22

Alternatively, you could use the equation for finding the mean. You could use x as the missing value.Then rearrange and solve.

Example 7: problem solving involving 2 means (HIGHER)

Data Set A has 6 values and a mean of 10 .

Data Set B has 4 values and a mean of 8 .

Find the mean of all 10 values.

For Set A – The mean of 6 values is 10 . We can multiply these together to find the total of the Set A numbers.

For Set B – The mean of 4 values is 8 . We can multiply these together to find the total of the Set B numbers.

We need to find the mean of all the values combined. We know there are 10 values. We can find the total of all 10 values by adding the total for Set A to the total of Set B .

We need to find the mean by dividing the total of all 10 values by 10 .

The mean of all the values is 9.2

Example 8: problem solving involving 2 means (HIGHER)

Data Set A has 4 values and a mean of 8 .

Data Set B has 7 values and a mean of 6 .

Find the mean of the all 11 values. Give your answer correct to 2 decimal places.  

For Set A – The mean of 4 values is 8 . We can multiply these together to find the total of the Set A numbers.

For Set B – The mean of 7 values is 6 . We can multiply these together to find the total of the Set B numbers.

We need to find the mean of all the values combined. We know there are 11 values. We can find the total of all 11 values by adding the total for Set A to the total of Set B .

We need to find the mean by dividing the total of all 11 values by 11 .

The mean of all the values is 6.73 \text{ (2 dp)}

Common misconceptions

• Check which average you are being asked for

Check if you have been asked for the median, mode, or mean average.

• The mean average can be a decimal

The mean does not have to be a whole number. It can be a decimal or a fraction. It may be a decimal which needs rounding.

Practice mean maths questions

1. Find the mean of this list of values: 5 7 8 8 9

2. Find the mean of this list of values: 3 5 6 9

3. Find the mean of this list of values. Give your answer to 3 significant figures: 7 8 9 10 10 11

4. Find the mean of this list of values. Give your answer to 3 significant figures: 17 18 18 19 20 21 25

5. The mean of 4 numbers is 9 . Here are 3 of the numbers 6 8 15 What is the 4 th number?

The total of 4 numbers is: \text{Total of 4 values}=\text{mean} \times \text{number of values}=9\times 4=36

The total of 3 numbers is: 6+8+9=29

The difference between the totals is: 36-29=7

This is the missing number.

6. The mean of 6 numbers is 12 . Here are 5 of the numbers 7 9 11 13 18 What is the 6 th number?

The total of 4 numbers is: \text{Total of 4 values}=\text{mean} \times \text{number of values}=12\times 6=72

The total of 5 numbers is: 7+9+11+13+18=58

The difference between the totals is: 72-58=14

Mean maths GCSE questions

\frac{1+3+4+6+6}{5}=\frac{20}{5} For adding the numbers and dividing by 5

=5 For the correct answer

\frac{15+15+19+20+54+59}{6}=\frac{182}{6} For adding the numbers and dividing by 6

\frac{182}{6}=30.333…=30.3 \ \text{(to 3 sf)} For the correct answer

5\times 61-(48+59+65+73) For subtracting the total of 4 results from the total of 5 results

305-245=60 For the correct answer

4. HIGHER 5 red bricks have a mean weight of 10 kg . 3 blue bricks have a mean weight of 6 kg . Find the mean of the 8 bricks.

For finding the total weights of the red bricks and the blue bricks

\frac{50+18}{8}=\frac{68}{8} For dividing the total weight of 8 bricks by 8

=8.5 \ \text{kg} For the correct answer

Learning checklist

You have now learned how to:

• Calculate the mean
• Solve problems involving the mean
• Higher – calculate the combined mean

The next lessons are

• Representing data
• Frequency table

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

Find out more about our GCSE maths tuition programme.

Privacy Overview

Word Problems: When to Use Mean, Median, and Mode - Expii

Word problems: when to use mean, median, and mode, explanations (4).

(Video) Word Problems: When to Use Mean, Median, and Mode

by Anusha Rahman

This video by Anusha Rahman goes over three examples of when to use mean, median, and mode in word problems.

Example 1: Jessie’s math teacher said that their final class grade is based on the average of all exam grades. Jessie has gotten a 93,87,71, and 97 on her math exams. What measure of central tendency will she use to calculate the average? What is the average?

We know that when we see the word average , we're talking about the mean ! So, we will be using mean to solve this problem.

Mean=93+87+71+974 =3484 =87

So, we know that Jessie's average score on her math exams was 87!

Example 2: Meghan wants to get an accurate picture of household salary in her neighborhood. The salaries for people in her neighborhood are: 87,000;94,000;103,000;97,000; and 55,000. What measure of central tendency will most accurate describe the data?

At first glance, we can see that there is an outlier at 55,000. At outlier is going to mess with the mean, and mode (the frequency) is not going to give us a lot of information. So, we will use the median, or the middle value.

Step 1: Put the salaries in numerical order, from least to greatest.

55,000,87,000,94,000,97,000,103,000

Step 2: Cross of the numbers evenly from each side until you're left with a middle number.

In this case, the median is 94,000.

Example 3: At her birthday party, Charlotte asks everyone to go around and say their favorite Taylor Swift song. In order, people respond with:

All Too Well, Shake it Off, Black Space, New Romantics, All Too Well, New Romantics, All Too Well, and Mr Perfectly Fine

Charlotte wants to know which song is the most popular. What measure of central tendency should she use?

Because the question is asking for the most popular song, we know we are looking for the mode ! We also know that the mean and median cannot be used for nominal or qualitative data.

Our first step is to make a frequency table with the data.

SongFrequencyAll Too Well3Shake It Off1Blank Space1New Romantics2Mr. Perfectly Fine1

From this frequency table, we can see that All Too Well has the highest frequency, so that is the mode of the data set, and it is the most popular song!

When solving word problems with central tendency, it's important to understand the pros and cons of each type . Let's break them down.

PRO: asymmetrical data (with an outlier)

CON: doesn't work with qualitative or nominal data

PRO: symmetrical data (no outlier)

PRO: big data sets

CON: highly affected by outliers

PRO: working with qualitative or nominal data

PRO: very easy to look at and read graphs/charts

CON: very limited in what it tells you

Related Lessons

(video) understanding and calculating measures of central tendency.

by Nurse Killam

This video by Nurse Killam explains when to use mean, median, and mode.

She starts with the definition of a normal distribution . All of the measures of central tendency work best with a normal distribution.

The mean is the average of a data set. We find the mean by adding all of the values, and dividing the sum by the number of values. The mean gets more accurate with more values.

Mean Benefits

• Best measure for symmetrical distributions
• Influenced by all data
• Most reliable
• Good for interval and ratio data

Mean Limitations

• Works best with no outliers

The median is the middle value of a set. We find this by ordering the data, then finding the literal middle number. Even sized sets will have two numbers that share the middle. The median of these sets is the average of those two numbers.

Median Benefits

• Good for asymmetrical data
• Works for ordinal, interval, and ratio data

Median Limitations

• Does not account for extreme scores
• Not algebraically defined
• Not appropriate for nominal data

The mode is the value that occurs the most in a set. We find this by ordering the set and finding the number that repeats the most.

Mode Benefits

• The only appropriate measure for nominal data
• Can also be used for ordinal, interval, and ratio data

Mode Limitations

• Cannot be used if all scores are different
• There can be several modes
• Cannot be used for further calculations

Using the Measures of Central Tendency: Media, Median, and Mode

When you're faced with word problems where you're asking to look into the distribution of a dataset, it can feel intimidating to figure out whether to use mean, median, or mode. The good news is that because each measure is different, they each have their own advantages and disadvantages!

Image source: by Anusha Rahman

Let's go through some example problems.

Which measure of central tendency is the most effective way to analyze the following dataset? 4,7,9,10,11,18,2,20,15,350,14

When to Use Mean, Median, and Mode

Three extremely important measures of central tendency (part of statistics ) are mean , median , and mode . You can find the mean, median, and mode of most data sets. Sometimes, it is more advantageous to only focus on one or two of these measures of central tendency, depending on your data set. Let's find out when we should use the mean, median, or mode to accurately describe a data set.

The mean is the average value in a data set. We can calculate the mean using the following formula:

mean=sum of all valuestotal number of values

Advantages:

The mean is the best measure for symmetrical data distributions. It is affected by all of the data points, and the more data points you know, the more accurate the mean will be. The mean is helpful when working with interval and ratio data.

Limitations:

Outliers can really throw off the mean. For example, say we have the data set {1,4,5,6,6,7,8,8,9}. The average of this data set is 1+4+5+6+6+7+8+8+99=549=6. But, what if we had an outlier in the set? What if our new set was {1,4,5,6,6,7,8,8,9,4946}? When our new mean would be 1+4+5+6+6+7+8+8+9+494610=500010=500. See how much an outlier can throw off the mean?

The median is the value separating the first half of the data set with the second half. To find the median of a data set, first put the data set in order. Then, if there are an odd number of values, cross an equal amount of values off from the beginning of the list and the end of the list. The remaining data point in the middle of your list is the median! If there are an even number of values in your set, the median will be the average of the two middle values.

The median is helpful when describing asymmetrical data , and it also works for ordinal, ratio, or interval data. It is easy to find, especially if your data is in order. Also, it is not usually too affected by outliers.

Limitations

However, the fact that the median is not usually affected by outliers means that we can't use it to account for these extreme values. Also, the median is not helpful when we are working with nominal or qualitative data. Think about it ... how can you order a list of colors and find the middle color?

The mode of a data set is the value that occurs most frequently. We can find the mode by putting the data in order, and then identifying which value(s) repeat the most. Frequency tables , histograms , and frequency distributions can be helpful in identifying the mode.

Finding the mode can be especially helpful when we are working with qualitative data or nominal data , but it can also be used when we have ordinal, ratio, or interval data. It's easy to spot on a graph, frequency table, or even an ordered set of data. It is not usually affected by outliers.

There can be many modes, so finding the mode may not say much about how the data is distributed. On the other hand, if all the data points only occur once, there is no real mode. The mode is unstable, and can be easily influenced by certain changes in the data. Adding just one new data point can change the mode!

Word Problems That Involve The Mean (I) Worksheet

Related Topics & Worksheets: Mean Worksheet 1 Mean Worksheet 3 More Statistics Worksheets

Objective: I know how to solve word problems that involve the mean. In statistics, mode, median and mean are typical values to represent a pool of numerical observations. They are calculated from the pool of observations. Mode is the most common value among the given observations. For example, a person who sells ice creams might want to know which flavor is the most popular. Median is the middle value, dividing the number of data into 2 halves. In other words, 50% of the observations is below the median and 50% of the observations is above the median. Mean is the average of all the values. For example, a teacher may want to know the average marks of a test in his class.

• When a number x is added to the set of numbers 5, 7, 8, 12, 6, 10 and 11; the new mean is 9. Calculate the value of x. x = 2) The mean height of the 30 children in a room is 76 cm. James, whose height is 107 cm went into the room. What is the new mean of the height of the children in the room? Mean height = cm 3) The mean of a set of 10 numbers is 60. When a number x is added to the set, the mean is changed to 65. Find the value of x. x = 4) The mean of a set of 12 numbers is 65. When a number y is removed from the set, the mean is changed to 63. Find the value of y. y =
• The mean weight of a group of 15 athletes was 60 kg. Two athletes whose weights were 68kg and 65 kg left the group. What is the new mean weight of the remaining athletes in the group? Mean weight = kg

 Check   Hint 

We hope that the free math worksheets have been helpful. We encourage parents and teachers to select the topics according to the needs of the child. For more difficult questions, the child may be encouraged to work out the problem on a piece of paper before entering the solution. We hope that the kids will also love the fun stuff and puzzles.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• How to Find the Mean | Definition, Examples & Calculator

How to Find the Mean | Definition, Examples & Calculator

Published on October 9, 2020 by Pritha Bhandari . Revised on June 21, 2023.

The mean (aka the arithmetic mean, different from the geometric mean ) of a dataset is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency and is often referred to as the “average.”

Table of contents

Arithmetic mean calculator, mean formulas for populations and samples, steps for calculating the mean by hand, outlier effect on the mean, when can you use the mean, median or mode, other interesting articles, frequently asked questions about the mean, here's why students love scribbr's proofreading services.

Discover proofreading & editing

In research, you often collect data from samples and perform inferential statistics to understand the population they came from.

The formulas for the sample mean and the population mean only differ in mathematical notation. Population attributes use capital letters while sample attributes use lowercase letters.

Population mean

The population mean can also be denoted as μ.

Sample mean

The sample mean is also referred to as M .

There are two steps for calculating the mean:

• Add up all the values in the data set.
• Divide this number by the number of values.

We’ll walk through these steps with a sample data set.

Let’s say you want to find the average amount people spend on a restaurant meal in your neighborhood. You ask a sample of 8 neighbors how much they spent the last time they went out for dinner, and find the mean cost.

Step 1: Find the sum of the values by adding them all up

Because we’re working with a sample, we use the sample formula.

Step 2: Divide the sum by the number of values

In the formula,  n is the number of values in your data set. Our data set has 8 values.

The mean tells us that in our sample, participants spent an average of 50 USD on their restaurant bill.

Outliers are extreme values that differ from most values in the data set. Because all values are used in the calculation of the mean, an outlier can have a dramatic effect on the mean by pulling the mean away from the majority of the values.

Let’s see what happens to the mean when we add an outlier to our data set.

As we can see, adding just one outlier to our data set raised the mean by 20 USD. In this case, a different measure of central tendency, like the median , would be more appropriate.

The mean is the most widely used measure of central tendency because it uses all values in its calculation. The best measure of central tendency depends on your type of variable and the shape of your distribution.

Type of variable

The mean can only be calculated for quantitative variables (e.g., height), and it can’t be found for categorical variables (e.g., gender).

In categorical variables, data is placed into groupings without exact numerical values, so the mean cannot be calculated. For categorical variables, the mode is the best measure of central tendency because it tells you the most common characteristic or popular choice for your sample.

But for continuous or discrete variables, you have exact numerical values. With these, you can easily calculate the mean or median.

Distribution shape

The mean is best for data sets with normal distributions . In a normal distribution, data is symmetrically distributed with no skew . Most values cluster around a central region, with values tapering off as they go further away from the center.

The mean, mode and median are exactly the same in a normal distribution.

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other.

For skewed distributions and distributions with outliers, the mean is easily influenced by extreme values and may not accurately represent the central tendency. The median is a better measure for these distributions as it takes a value from the middle of the distribution.

Alternatively, you can systematically review and remove outliers from your dataset in the data cleansing process.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Statistical power
• Pearson correlation
• Degrees of freedom
• Statistical significance

Methodology

• Cluster sampling
• Stratified sampling
• Focus group
• Systematic review
• Ethnography
• Double-Barreled Question

Research bias

• Implicit bias
• Publication bias
• Cognitive bias
• Placebo effect
• Pygmalion effect
• Hindsight bias
• Overconfidence bias

You can find the mean , or average, of a data set in two simple steps:

• Find the sum of the values by adding them all up.
• Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.

The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

• Weighted mean: some values contribute more to the mean than others.
• Geometric mean : values are multiplied rather than summed up.
• Harmonic mean: reciprocals of values are used instead of the values themselves.

Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

• The mode is the most frequent value.
• The median is the middle number in an ordered data set.
• The mean is the sum of all values divided by the total number of values.

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bhandari, P. (2023, June 21). How to Find the Mean | Definition, Examples & Calculator. Scribbr. Retrieved April 9, 2024, from https://www.scribbr.com/statistics/mean/

Is this article helpful?

Pritha Bhandari

Other students also liked, how to find the median | definition, examples & calculator, how to find the mode | definition, examples & calculator, how to calculate standard deviation (guide) | calculator & examples, what is your plagiarism score.

How to Find the Mean

The mean is the average of the numbers.

It is easy to calculate: add up all the numbers, then divide by how many numbers there are.

In other words it is the sum divided by the count .

Example 1: What is the Mean of these numbers?

• Add the numbers: 6 + 11 + 7 = 24
• Divide by how many numbers (there are 3 numbers): 24 / 3 = 8

The Mean is 8

Why does this work.

It is because 6, 11 and 7 added together is the same as 3 lots of 8:

It is like you are "flattening out" the numbers

Example 2: Look at these numbers:

3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29

The sum of these numbers is 330

There are fifteen numbers.

The mean is equal to 330 / 15 = 22

The mean of the above numbers is 22

Negative numbers.

How do you handle negative numbers? Adding a negative number is the same as subtracting the number (without the negative). For example 3 + (−2) = 3−2 = 1.

Knowing this, let us try an example:

Example 3: Find the mean of these numbers:

3, −7, 5, 13, −2

• The sum of these numbers is 3 − 7 + 5 + 13 − 2 = 12
• There are 5 numbers.
• The mean is equal to 12 ÷ 5 = 2.4

The mean of the above numbers is 2.4

Here is how to do it one line:

Mean = 3 − 7 + 5 + 13 − 2 5 = 12 5 = 2.4

Try it yourself!

Now have a look at The Mean Machine .

Advanced Topic: the mean we have just looked at is also called the Arithmetic Mean , because there are other means such as the Geometric Mean and Harmonic Mean .

• Varsity Tutors
• K-5 Subjects
• Study Skills
• All AP Subjects
• AP Calculus
• AP Chemistry
• AP Computer Science
• AP Human Geography
• AP Macroeconomics
• AP Microeconomics
• AP Statistics
• AP US History
• AP World History
• All Business
• Business Calculus
• Microsoft Excel
• Supply Chain Management
• All Humanities
• Essay Editing
• All Languages
• Mandarin Chinese
• Portuguese Chinese
• Sign Language
• All Learning Differences
• Learning Disabilities
• Special Education
• College Math
• Common Core Math
• Elementary School Math
• High School Math
• Middle School Math
• Pre-Calculus
• Trigonometry
• All Science
• Organic Chemistry
• Physical Chemistry
• All Engineering
• Chemical Engineering
• Civil Engineering
• Computer Science
• Electrical Engineering
• Industrial Engineering
• Materials Science & Engineering
• Mechanical Engineering
• Thermodynamics
• Biostatistics
• College Essays
• High School
• College & Adult
• 1-on-1 Private Tutoring
• Online Tutoring
• Instant Tutoring
• Pricing Info
• All AP Exams
• ACT Tutoring
• ACT Reading
• ACT Science
• ACT Writing
• SAT Tutoring
• SAT Reading
• SAT Writing
• GRE Tutoring
• NCLEX Tutoring
• Real Estate License
• And more...
• StarCourses
• Beginners Coding
• Early Childhood
• For Schools Overview
• High-Dosage Tutoring
• Free 24/7 Tutoring & Classes
• Learning Outcomes
• About The Tutors
• Talk with Our Team
• Reviews & Testimonials
• Press & Media Coverage
• Tutor/Instructor Jobs
• Corporate Solutions
• About Nerdy
• Become a Tutor

Word Problems Involving the Mean of a Data Set

As we delve deeper into the world of statistics and probability, one specific term becomes very important to us: The mean. You've probably encountered this concept before -- even if you weren't aware of it. "Mean" essentially means the same thing as "average," and this number becomes very useful in a number of different fields. A baseball player's "batting average" tells you how skilled he or she is. The "average" return of an investment helps us understand how profitable it really is. As we can see, the mean is very useful in real-world applications. Let's tackle some word problems that show us how we can apply means to these real-world situations.

What is the mean?

We can find the mean of any data set with two easy steps:

• Add all the data points together
• Divide by the number of data points

Solving word problems involving means

Now let's put our knowledge to good use and solve a few word problems that involve means:

Consider the following data set: 4 , 8 , 20 , 25 , 32

If we add one more number ( x ) to this data set, we are left with a mean of 15. What is the value of this additional number?

First, let's try to add all of our numbers together: 4 + 8 + 20 + 25 + 32 + x

We don't know the value of x, so we can't complete this addition. But let's continue with the operation anyway:

If we add one more value (x) to our current data set, we get 6 total data points. This means that we need to divide the above sum by six:

4 + 8 + 20 + 25 + 32 + x + 6

We know that the end result is a mean of 15. So how can we solve this equation? Easy: We simplify it.

4 + 8 + 20 + 25 + 32 + x + 6 = 15

89 + x / 6 = 15

Now, all we need to do is multiply each side of the equation by 6:

6 × ( 89 + x ) / ( 6 ) = 15 × 6

89 + x = 90

The answer is 1.

We can even use our knowledge of means to help us set higher goals!

What happens if we know that we need an average of 90% or higher to get an "A" in math class?

What if our past test scores were 80%, 85%, 88%, and 95%?

We have one more test for the term. What score do we need to get an "A" average?

Right away, we know that our end result needs to be at least 90.

We also know that the total number of tests is 5.

With these values, we can find our minimum test score for an A average (x).

80 + 85 + 88 + 95 + x / 5

We can simplify this as 348 + x / 5

Now, all we need to do is multiply each side of the equation by 5:

5 × ( 348 + x ) / ( 5 ) = 90 × 5

Or: 348 + x = 450

Uh oh, looks like that value is 102. As we know, the highest possible score on a test is 100%. This means that it's theoretically impossible for us to get an A average this term -- unless, of course, we speak to our teacher about the possibility of doing additional work for extra credit. This highlights the importance of calculating means in order to plan effectively for the future.

Exercises such as this also show us how even a few low scores can seriously affect the overall average or mean.

The especially low score of 80% is 10 percentage points below our target grade, and this skews the data toward a lower mean. We can consider this value an outlier.

Topics related to the Word Problems Involving the Mean of a Data Set

Center and Variation of Data

Word Problems

Interpreting Data

Flashcards covering the Word Problems Involving the Mean of a Data Set

Statistics Flashcards

Common Core: High School - Statistics and Probability Flashcards

Practice tests covering the Word Problems Involving the Mean of a Data Set

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

Tutoring helps your student gain confidence when solving word problems involving means

Tutoring gives your student the opportunity to go over a few more examples of word problems involving means. Sometimes, all it takes to reach for confidence in these concepts is a little extra time outside of class. Tutors can even cater to your student's learning style during these sessions, whether they prefer verbal or visual cues. Speak with Varsity Tutors' Educational Directors today to learn more about the possibilities of tutoring. We'll match your student with a suitable tutor.

Download our free learning tools apps and test prep books

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Course: 6th grade   >   Unit 11

• Statistics intro: Mean, median, & mode
• Mean, median, & mode example

Calculating the mean

• Calculating the median
• Calculating the mean: data displays
• Calculating the median: data displays
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

• Math Article

Mean Questions

In mathematics and statistics, the mean is one of the important concepts that can be understood easily without much prior knowledge. Different types of the mean are defined based on the data provided. In this article, you will get questions and solutions for finding the mean of a given data set and grouped data (frequency distribution). These questions will help the students of Classes 9 and 10 to get enough practice for this concept.

What is Mean in Statistics?

Mean is one of the measures of central tendency in statistics. The mean is the average of the given data set, which means it can be calculated by dividing the sum of the given data values by the total number of data values.

Mean for ungrouped data:

Mean (x̄) = ∑x i /n

x i = x 1 , x 2 , x 3 ,…, x n such that i = 1, 2, 3,…n.

Number of observations = n

Mean for grouped data:

Mean (x̄) = ∑f i x i / ∑f i

Here, f i ’s are frequencies of x i ’s.

Mean Questions and Answers

1. Calculate the mean from the data showing marks of students in a class in a test: 40, 50, 55, 78, 58.

Given marks:

40, 50, 55, 78, 58

Here, the number of data values = 5

We know that:

Mean = Sum of data values/Total number of data values

= (40 + 50 + 55 + 78 + 58)/5

Therefore, the mean for the given data is 56.2.

2. A class consists of 50 students, out of which 30 are girls. The mean of marks scored by girls in a test is 73 (out of 100), and that of boys is 71. Determine the mean score of the whole class.

Total number of students in a class = 50

Number of girls in the class = 30

Number of boys in the class = 50 – 30 = 20

Mean marks scored by girls = 73

Mean marks scored by boys = 71

Thus, the total marks scored by girls = 73 × 30 = 2190

Also, the total marks scored by boys = 71 × 20 = 1420

Mean score of the class = (Total marks scored by girls and boys)/Total number of students

= (2190 + 1420)/50

3. The mean of the following distribution is 50.

Find the value of a and hence the frequencies of 30 and 70.

We know that,

Mean for grouped data (x̄) = ∑f i x i /n

(640a + 2800)/ (12a + 60) = 50 {given that mean = 50}

640a + 2800 = 50(12a + 60)

640a + 2800 = 600a + 3000

640a – 600a = 3000 – 2800

Therefore, the frequency for 30 = 5a + 3 = 5(5) + 3 = 28

And the frequency for 70 = 7a – 11 = 7(5) – 11 = 24

4. Find the mean salary of 60 workers of a factory from the following table:

Mean = (305000)/60 = 5083.33.

Therefore, the mean salary = Rs. 5083.33

5. A total of 25 patients admitted to a hospital are tested for levels of blood sugar, (mg/dl) and the results obtained were as follows:

87, 71, 83, 67, 85, 77, 69, 76, 65, 85, 85, 54, 70, 68, 80, 73, 78, 68, 85, 73, 81, 78, 81, 77, 75

Find the mean (mg/dl) of the above data.

Sum of data values = 87 + 71 + 83 + 67 + 85 + 77 + 69 + 76 + 65 + 85 + 85 + 54 + 70 + 68 + 80 + 73 + 78 + 68 + 85 + 73 + 81 + 78 + 81 + 77 + 75

Mean = 1891/25

6. Calculate the mean for the following distribution.

Mean = a + (∑f i d i /∑f i )

= 47.5 + (435/30)

= 47.5 + 14.5

7. The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify?

The given distribution has unequal class heights. So, we can use the direct method for calculating the mean.

Mean = ∑f i x i / ∑f i

Here the mean signifies that, on average, 45 bowlers take 152.89 wickets.

8. A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Which method did you use for finding the mean, and why?

Here, the values of fi and xi are small. Thus, we can use the direct method to calculate the mean.

So, Mean = ∑f i x i / ∑f i

Therefore, the mean number of plants = 8.1

9. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency x.

Mean pocket allowance = a + (∑f i d i / ∑f i )

18 + [(2x – 40)/(44 + x)] = 18 [from the given]

(2x – 40)/(44 + x) = 18 – 18

(2x – 40)/(44 + x) = 0

2x – 40 = 0

Hence, the missing frequency is 20.

10. The following is the cumulative frequency distribution (of less than type) of 1000 persons, each one of age 20 years and above. Determine the mean age.

For the given frequency distribution, we need to write the class intervals and frequencies and then proceed with the necessary calculations for estimating the mean age.

Mean = a + h(∑f i u i /∑f i )

= 55 + 10(-370/1000)

= 55 – 3.7

Therefore, the mean age is 51.3 years.

Practice Questions on Mean

• The points scored by a basketball team in a series of matches are as follows: 17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 48, 10, 8, 7, 10, 28

Find the mean for the data.

• 5 people were asked about the time in a week they spent doing social work in their community. They said 10, 7, 13, 20 and 15 hours, respectively. Find the mean (or average) time in a week devoted by them to social work.
• The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.
• The frequency distribution table of agricultural holdings in a village is given below.

Find the mean agricultural holdings of the village.

• Calculate the mean for the following distribution.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

The Mean Value Theorem is one of the most important theorems in Introductory Calculus, and it forms the basis for proofs of many results in subsequent and advanced Mathematics courses. The history of this theorem begins in the 1300's with the Indian Mathematician Parameshvara , and is eventually based on the academic work of Mathematicians Michel Rolle in 1691 and Augustin Louis Cauchy in 1823. The formal statement of this theorem together with an illustration of the theorem will follow. I will also state Rolle's Theorem , which is used in the proof the Mean Value Theorem. Both theorems are given without proof, and all subsequent problems here will be referencing only the Mean Value Theorem. All functions are assumed to be real-valued. ROLLE'S THEOREM: Let $f$ be a continuous function on the closed interval $ [a, b] $ and differentiable on the open interval $ (a, b) $ with $f(a)=f(b)$. Then there is at least one number $c$ ($x$-value) in the interval $(a, b)$ which satifies $$ f'(c)=0 $$ MEAN VALUE THEOREM: Let $f$ be a continuous function on the closed interval $ [a, b] $ and differentiable on the open interval $ (a, b) $. Then there is at least one number $c$ ($x$-value) in the interval $(a, b)$ which satifies $$ f'(c)= \displaystyle{ f(b) - f(a) \over b-a } $$ Numerous proofs for Rolle's Theorem and the Mean Value Theorem can easily be found on the internet. I have attached proofs of both Theorems here , along with other results related to the Mean-Value Theorem. In the list of Mean Value Theorem Problems which follows, most problems are average and a few are somewhat challenging. PROBLEM 1 : Determine if the Mean Value Theorem can be applied to the following function on the the given closed interval. If so, find all possible values of $c$: $ \ f(x)=3+ \sqrt{x} \ $ on $ \ [0, 4] $ Click HERE to see a detailed solution to problem 1.

How it works

For Business

Join Mind Tools

Article • 7 min read

What Is Problem Solving?

By the Mind Tools Content Team

We all spend a lot of our time solving problems, both at work and in our personal lives.

Some problems are small, and we can quickly sort them out ourselves. But others are complex challenges that take collaboration, creativity, and a considerable amount of effort to solve.

At work, the types of problems we face depend largely on the organizations we're in and the jobs we do. A manager in a cleaning company, for example, might spend their day untangling staffing issues, resolving client complaints, and sorting out problems with equipment and supplies. An aircraft designer, on the other hand, might be grappling with a problem about aerodynamics, or trying to work out why a new safety feature isn't working. Meanwhile, a politician might be exploring solutions to racial injustice or climate change.

But whatever issues we face, there are some common ways to tackle them effectively. And we can all boost our confidence and ability to succeed by building a strong set of problem-solving skills.

Mind Tools offers a large collection of resources to help you do just that!

How Well Do You Solve Problems?

Start by taking an honest look at your existing skills. What's your current approach to solving problems, and how well is it working? Our quiz, How Good Is Your Problem Solving? lets you analyze your abilities, and signposts ways to address any areas of weakness.

Define Every Problem

The first step in solving a problem is understanding what that problem actually is. You need to be sure that you're dealing with the real problem – not its symptoms. For example, if performance in your department is substandard, you might think that the problem lies with the individuals submitting work. However, if you look a bit deeper, the real issue might be a general lack of training, or an unreasonable workload across the team.

Tools like 5 Whys , Appreciation and Root Cause Analysis get you asking the right questions, and help you to work through the layers of a problem to uncover what's really going on.

However, defining a problem doesn't mean deciding how to solve it straightaway. It's important to look at the issue from a variety of perspectives. If you commit yourself too early, you can end up with a short-sighted solution. The CATWOE checklist provides a powerful reminder to look at many elements that may contribute to the problem, keeping you open to a variety of possible solutions.

Understanding Complexity

As you define your problem, you'll often discover just how complicated it is. There are likely several interrelated issues involved. That's why it's important to have ways to visualize, simplify and make sense of this tangled mess!

Affinity Diagrams are great for organizing many different pieces of information into common themes, and for understanding the relationships between them.

Another popular tool is the Cause-and-Effect Diagram . To generate viable solutions, you need a solid understanding of what's causing the problem.

When your problem occurs within a business process, creating a Flow Chart , Swim Lane Diagram or a Systems Diagram will help you to see how various activities and inputs fit together. This may well highlight a missing element or bottleneck that's causing your problem.

Quite often, what seems to be a single problem turns out to be a whole series of problems. The Drill Down technique prompts you to split your problem into smaller, more manageable parts.

General Problem-Solving Tools

When you understand the problem in front of you, you’re ready to start solving it. With your definition to guide you, you can generate several possible solutions, choose the best one, then put it into action. That's the four-step approach at the heart of good problem solving.

There are various problem-solving styles to use. For example:

• Constructive Controversy is a way of widening perspectives and energizing discussions.
• Inductive Reasoning makes the most of people’s experiences and know-how, and can speed up solution finding.
• Means-End Analysis can bring extra clarity to your thinking, and kick-start the process of implementing solutions.

Specific Problem-Solving Systems

Some particularly complicated or important problems call for a more comprehensive process. Again, Mind Tools has a range of approaches to try, including:

• Simplex , which involves an eight-stage process: problem finding, fact finding, defining the problem, idea finding, selecting and evaluating, planning, selling the idea, and acting. These steps build upon the basic, four-step process described above, and they create a cycle of problem finding and solving that will continually improve your organization.
• Appreciative Inquiry , which is a uniquely positive way of solving problems by examining what's working well in the areas surrounding them.
• Soft Systems Methodology , which takes you through four stages to uncover more details about what's creating your problem, and then define actions that will improve the situation.

Further Problem-Solving Strategies

Good problem solving requires a number of other skills – all of which are covered by Mind Tools.

For example, we have a large section of resources to improve your Creativity , so that you come up with a range of possible solutions.

By strengthening your Decision Making , you'll be better at evaluating the options, selecting the best ones, then choosing how to implement them.

And our Project Management collection has valuable advice for strengthening the whole problem-solving process. The resources there will help you to make effective changes – and then keep them working long term.

Problems are an inescapable part of life, both in and out of work. So we can all benefit from having strong problem-solving skills.

It's important to understand your current approach to problem solving, and to know where and how to improve.

Define every problem you encounter – and understand its complexity, rather than trying to solve it too soon.

There's a range of general problem-solving approaches, helping you to generate possible answers, choose the best ones, and then implement your solution.

Some complicated or serious problems require more specific problem-solving systems, especially when they relate to business processes.

By boosting your creativity, decision-making and project-management skills, you’ll become even better at solving all the problems you face.

You've accessed 1 of your 2 free resources.

Get unlimited access

Discover more content

Book Insights

The Back of the Napkin: Solving Problems and Selling Ideas With Pictures

Infographic

Creative Problem Solving Infographic

Infographic Transcript

Add comment

Comments (0)

Be the first to comment!

Team Management

Learn the key aspects of managing a team, from building and developing your team, to working with different types of teams, and troubleshooting common problems.

Sign-up to our newsletter

Subscribing to the Mind Tools newsletter will keep you up-to-date with our latest updates and newest resources.

Subscribe now

Business Skills

Personal Development

Leadership and Management

Member Extras

Most Popular

Newest Releases

Pain Points Podcast - Connecting to Mission

Empathy at Work

Mind Tools Store

About Mind Tools Content

Discover something new today

The five conversations framework.

An Alternative Approach to Appraisals

What Is the Five Conversations Framework?

Changing the way you do appraisals

How Emotionally Intelligent Are You?

Boosting Your People Skills

Self-Assessment

What's Your Leadership Style?

Learn About the Strengths and Weaknesses of the Way You Like to Lead

Recommended for you

Are you intrinsically motivated.

Understanding What Drives You

Business Operations and Process Management

Strategy Tools

Customer Service

Business Ethics and Values

Handling Information and Data

Project Management

Knowledge Management

Self-Development and Goal Setting

Time Management

Presentation Skills

Learning Skills

Career Skills

Communication Skills

Negotiation, Persuasion and Influence

Working With Others

Difficult Conversations

Creativity Tools

Self-Management

Work-Life Balance

Stress Management and Wellbeing

Coaching and Mentoring

Change Management

Managing Conflict

Delegation and Empowerment

Performance Management

Leadership Skills

Developing Your Team

Talent Management

Problem Solving

Decision Making

Member Podcast

• Bipolar Disorder
• Therapy Center
• When To See a Therapist
• Types of Therapy
• Best Online Therapy
• Best Couples Therapy
• Best Family Therapy
• Managing Stress
• Sleep and Dreaming
• Understanding Emotions
• Self-Improvement
• Healthy Relationships
• Student Resources
• Personality Types
• Guided Meditations
• Verywell Mind Insights
• 2023 Verywell Mind 25
• Mental Health in the Classroom
• Editorial Process
• Meet Our Review Board
• Crisis Support

Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

JGI / Jamie Grill / Getty Images

• Application
• Improvement

From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261

Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality .  Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition .  Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568

National Alliance on Mental Illness. Warning signs and symptoms .

Mayer RE. Thinking, problem solving, cognition, 2nd ed .

Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

• Home   |  
• About   |  
• Contact Us   |  
• Privacy   |  
• Copyright   |  
• Shop   |  
• 🔍 Search Site
• Easter Color By Number Sheets
• Printable Easter Dot to Dot
• Easter Worksheets for kids
• Kindergarten
• All Generated Sheets
• Place Value Generated Sheets
• Addition Generated Sheets
• Subtraction Generated Sheets
• Multiplication Generated Sheets
• Division Generated Sheets
• Money Generated Sheets
• Negative Numbers Generated Sheets
• Fraction Generated Sheets
• Place Value Zones
• Number Bonds
• Addition & Subtraction
• Times Tables
• Fraction & Percent Zones
• All Calculators
• Fraction Calculators
• Percent calculators
• Area & Volume Calculators
• Age Calculator
• Height Calculator
• Roman Numeral Calculator
• Coloring Pages
• Fun Math Sheets
• Math Puzzles
• Mental Math Sheets
• Online Times Tables
• Online Addition & Subtraction
• Math Grab Packs
• All Math Quizzes
• 1st Grade Quizzes
• 2nd Grade Quizzes
• 3rd Grade Quizzes
• 4th Grade Quizzes
• 5th Grade Quizzes
• 6th Grade Math Quizzes
• Place Value
• Rounding Numbers
• Comparing Numbers
• Number Lines
• Prime Numbers
• Negative Numbers
• Roman Numerals
• Subtraction
• Add & Subtract
• Multiplication
• Fraction Worksheets
• Learning Fractions
• Fraction Printables
• Percent Worksheets & Help
• All Geometry
• 2d Shapes Worksheets
• 3d Shapes Worksheets
• Shape Properties
• Geometry Cheat Sheets
• Printable Shapes
• Coordinates
• Measurement
• Math Conversion
• Statistics Worksheets
• Bar Graph Worksheets
• Venn Diagrams
• All Word Problems
• Finding all possibilities
• Logic Problems
• Ratio Word Problems
• All UK Maths Sheets
• Year 1 Maths Worksheets
• Year 2 Maths Worksheets
• Year 3 Maths Worksheets
• Year 4 Maths Worksheets
• Year 5 Maths Worksheets
• Year 6 Maths Worksheets
• All AU Maths Sheets
• Kindergarten Maths Australia
• Year 1 Maths Australia
• Year 2 Maths Australia
• Year 3 Maths Australia
• Year 4 Maths Australia
• Year 5 Maths Australia
• Meet the Sallies
• Certificates

Mean Worksheets

Welcome to our Mean Worksheets page. Here you will find a wide range of free printable Worksheets, which will help your child learn how to find the mean of a range of numbers, and also how to find missing data points when the mean is given.

These worksheets are aimed at students in 5th and 6th grade.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Quicklinks to ...

What is the Mean?

• How to Find the Mean
• How to Use the Mean to find a Missing Data Point

Find the Mean Worksheets

Missing data point worksheets.

• More related resources

Find the Mean Online Quiz

The mean is the average of a set of numbers.

It is found by adding up the set of numbers and then dividing the total by the number of data points in the set.

How to find the mean

Step 1) Add up all the numbers in the set.

Step 2) Divide the total by the total number of data points in the set.

Example 1) Find the mean of 5, 7, 8 and 4

Step 1) Add up the numbers to give a total of 5+7+8+4=24

Step 2) Divide the total by the number of data points. 24 ÷ 4 = 6

Answer: the mean is 6.

Example 2) Find the mean of 8, 2, 5, 7 and 13

Step 1) Add up the numbers to give a total of 8+2+5+7+13=35

Step 2) Divide by the number of data points. 35 ÷ 5 = 7

Answer: the mean is 7.

We have split these worksheets into two sections:

• Use the Mean to Find the Missing Data Point Worksheets

These printable mean worksheets have been carefully graded to ensure a progression in the level of difficulty.

The first 2 sheets involve finding the mean of 3, 4 or 5 relatively small numbers.

The third sheet involves finding the mean of 3, 4 or 5 numbers up to 100 and includes examples with decimals and fractions (halves only).

The 4th sheet includes decimals and negative numbers as well as positive ones.

Sheets 1,2 and 3 are suitable for 5th grade and 6th grade.

Sheet 4 is designed for 6th grade and up.

• Find the Mean Sheet 1
• PDF version
• Find the Mean Sheet 2
• Find the Mean Sheet 3
• Find the Mean Sheet 4

How To Use the Mean to Find a Missing Data Point

The Missing Data Point worksheets involve finding a missing data point, when there is one data point missing, and the mean is given.

These sheets are more complex than simply finding the mean of a range of numbers.

How to find a missing data point

Step 2) Multiply the mean by the total number of data points in the set.

Step 3) The missing data point is equal to the difference between the answer in Step 2 and the answer in Step 1.

Example 1) The mean of 4 numbers is 6. Three of the numbers are 4, 8, and 3. What is the 4th number?

Step 1) Add up the 3 numbers to give a total of 4+8+3=15

Step 2) Multiply the mean by the number of data points. 6x4=24

Step 3) The missing data point is the answer from Step 2 subtract the answer from Step 1 which is 25-15=9.

Example 2) Five numbers have a mean of 3. Four of the numbers are 7, 5, 2 and -4. What is the 5th number?

Step 1) Add up the numbers to give a total of 7+5+2+(-4)=10

Step 2) Multiply the mean by the number of data points. 3x5=15

Step 3) The missing data point is the answer to Step 2 - Step 1, which is 15-10=5.

The first two sheets are suitable for 5th and 6th graders.

The third sheet is designed for 6th grade and up.

• Missing Data Point Sheet 1
• Missing Data Point Sheet 2
• Missing Data Point Sheet 3

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

• Median Worksheets

The sheets in this section will help you to find the median of a range of numbers, including negative numbers and decimals.

On some of the easier sheets, only odd numbers of data points have been used.

On the harder sheets, both odd and even numbers of data points have been included.

• Mode and Range Worksheets

The sheets in this section will help you to find the mode and range of a set of numbers, including negative numbers and decimals.

There are easier sheets involving fewer data points, and harder ones with more data points.

• Mean Median Mode and Range Worksheets

The sheets in this section will help you to find the mean, median, mode and range of a set of numbers, including negative numbers and decimals.

The sheets in this section will help you to solve problems involving bar graphs and picture graphs.

There are a range of sheet involving reading and interpreting graphs as well as drawing your own graphs.

Our quizzes have been created using Google Forms.

At the end of the quiz, you will get the chance to see your results by clicking 'See Score'.

This will take you to a new webpage where your results will be shown. You can print a copy of your results from this page, either as a pdf or as a paper copy.

For incorrect responses, we have added some helpful learning points to explain which answer was correct and why.

We do not collect any personal data from our quizzes, except in the 'First Name' and 'Group/Class' fields which are both optional and only used for teachers to identify students within their educational setting.

We also collect the results from the quizzes which we use to help us to develop our resources and give us insight into future resources to create.

For more information on the information we collect, please take a look at our Privacy Policy

We would be grateful for any feedback on our quizzes, please let us know using our Contact Us link, or use the Facebook Comments form at the bottom of the page.

This quick quiz tests your knowledge and skill at finding the mean and using the mean value to find missing data points..

How to Print or Save these sheets 🖶

Need help with printing or saving? Follow these 3 steps to get your worksheets printed perfectly!

• How to Print support

Return to 6th Grade Math Worksheets

Return to Statistics Hub page

Return from Mean Worksheets to Math Salamanders Homepage

Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.

New! Comments

TOP OF PAGE

© 2010-2024 Math Salamanders Limited. All Rights Reserved.

• Privacy Policy
• Copyright Policy

• International
• Schools directory
• Resources Jobs Schools directory News Search

Mean average - including problem solving and applying knowledge to different scenarios

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

22 February 2018

• Share through email
• Share through twitter
• Share through linkedin
• Share through facebook
• Share through pinterest

Creative Commons "Sharealike"

Your rating is required to reflect your happiness.

It's good to leave some feedback.

Something went wrong, please try again later.

Great set of questions, although in the major extension part C, the city B appears to have drifted 10 miles further away from A while they were there! Talk about plate movement!

Empty reply does not make any sense for the end user

101ashraffm

The questions and solutions was very useful, thank you.

Very useful; thank you!

Excellent resource, thank you for sharing! :)

Report this resource to let us know if it violates our terms and conditions. Our customer service team will review your report and will be in touch.

Not quite what you were looking for? Search by keyword to find the right resource:

All Interactive Maps and Locations

Sphinx Riddle Solutions and Locations

The sphinx is a rare creature in Dragon’s Dogma 2, and, as you’d expect, it has a few riddles for you to solve. This guide will help you find the sphinx and answer its convoluted questions correctly to earn several amazing rewards. Here are the sphinx’s location and riddle solutions in Dragon’s Dogma 2.

Where to Find the Sphinx

The sphinx can be found at the Mountain Shrine in northwest Vermund. Make your way through the Ancient Battleground to find the Worldsend Cavern. The cavern entrance is at the back of the castle behind a cyclops. You can ignore the cyclops and drake fighting in the field below.

Climb the nearby ladder after defeating the cyclops to find a path through the destroyed castle wall. The cavern entrance is much easier to see during the daytime, as there are zero light sources near the mouth of the cavern.

Continue through the linear Ancient Ruins and Shrine Corridor to reach the Mountain Shrine where the sphinx is located. Walk up the stairs once you exit the Shrine Corridor and you’ll find the sphinx waiting there. A conversation will automatically start as soon as you climb the stairs in front of the sphinx.

Sphinx Riddle Solutions at Mountain Shrine

The Sphinx asks five riddles in total at this shrine before moving on to the next one. Here are the riddles and answers for the conversation at the Mountain Shrine:

Where to Find the Second Sphinx Shrine

The sphinx will fly away after you solve all five riddles listed above, so you’ll need to find its next shrine to continue this questline. This time, the sphinx is waiting for you at the Frontier Shrine, and thankfully, this spot is much easier to find.

Go to the Checkpoint Rest Town and walk along the dried up ravine that runs underneath the town’s main entrance. You can see the path drawn lightly on the map above, and we’ve provided a clearer picture of where you should be in the screenshot below. You can see the town and the path that leads down, below it.

Eventually, the path leads you to a foggy area where you need to defeat a chimera and other large monsters to reach the sphinx. Once the area is clear, look for this large pillar and walk underneath it to find the Frontier Shrine. Be sure to open up the chests in the area before proceeding, too.

This is a much easier shrine to reach, but it’s important to drop a portable Portcrystal at the Frontier Shrine to help with the upcoming riddles – just like last time.

Sphinx Riddle Solutions at Frontier Shrine

There are five more riddles to solve at the Frontier Shrine, but as a word of warning – remember to switch to the Archer vocation before solving the final riddle. You need to use the Unmaking Arrow on the sphinx after solving the fifth riddle here.

How to Open the Sphinx’s Gold Chest

Even if you solve every single riddle asked, the sphinx won’t unlock the large, golden chest. The trick here – and perhaps the toughest riddle – is to defeat the sphinx in combat. You can buy a second Unmaking Arrow from the Dragonforged in Battahl to give yourself two chances here.

The game saves right after you shoot the arrow, so steady your aim. If you hit your shot, the sphinx will congratulate you and drop the Key of Sagacity . The contents of the gold chest are now yours to take. Inside, you’ll find the Eternal Wakestone , which revives all NPCs at once.

This is the only item that can save your playthrough from a ton of failed quests if you happen to let a pawn’s Dragonsplague reach the absolute limit.

Up Next: The Dragonforged

Top guide sections.

• Walkthrough
• Side Quests
• Boss Guides

Was this guide helpful?

Watch CBS News

Why is looking at a solar eclipse dangerous without special glasses? Eye doctors explain.

By Sara Moniuszko

Edited By Allison Elyse Gualtieri

Updated on: April 8, 2024 / 8:54 AM EDT / CBS News

The solar eclipse will be visible for millions of Americans on April 8, 2024, making many excited to see it — but how you watch it matters, since it can be dangerous for your eyes. 

A  solar eclipse occurs when the moon passes between the sun and Earth, blocking the sun's light . When the moon blocks some of the sun, it's a partial solar eclipse, but when moon lines up with the sun, blocking all of its light, a total solar eclipse occurs,  NASA explains . Either way, you need eye protection when viewing.

"The solar eclipse will be beautiful, so I hope that everyone experiences it — but they need to experience it in the right way," said Dr. Jason P. Brinton, an ophthalmologist and medical director at Brinton Vision in St. Louis.

Here's what to know to stay safe.

Why is looking at a solar eclipse dangerous?

Looking at the sun — even when it's partially covered like during an eclipse — can cause eye damage.

There is no safe dose of solar ultraviolet rays or infrared radiation, said  Dr. Yehia Hashad , an ophthalmologist, retinal specialist and the chief medical officer at eye health company Bausch + Lomb.

"A very small dose could cause harm to some people," he said. "That's why we say the partial eclipse could also be damaging. And that's why we protect our eyes with the partial as well as with the full sun."

Some say that during a total eclipse, it's safe to view the brief period time when the moon completely blocks the sun without eye protection. But experts warn against it. 

"Totality of the eclipse lasts only about 1 to 3 minutes based on geographic location, and bright sunlight suddenly can appear as the moon continues to move," notes an eclipse viewing guide published in JAMA , adding, "even a few seconds of viewing the sun during an eclipse" can temporarily or permanently damage your vision. 

Do I need special glasses for eclipse viewing?

Yes.  Eclipse glasses are needed to protect your eyes if you want to look at the eclipse.

Regular sunglasses aren't protective enough for eclipse viewing — even if you stack more than one. 

"There's no amount of sunglasses that people can put on that will make up for the filtering that the ISO standard filters and the eclipse glasses provide," Brinton said.

You also shouldn't look at the eclipse through a camera lens, phone, binoculars or telescope, according to NASA, even while wearing eclipse glasses. The solar rays can burn through the lens and cause serious eye injury.

Eclipse glasses must comply with the  ISO 12312-2 international safety standard , according to NASA, and should have an "ISO" label printed on them to show they comply. The American Astronomical Society  has a list  of approved solar viewers.

Can't find these, or they're sold out near you? You can also  make homemade viewers ,   which allow you to observe the eclipse indirectly — just don't accidentally look at the sun while using one.

How to keep kids safe during the solar eclipse

Since this eclipse is expected to occur around the time of dismissal for many schools across the country, it may be tempting for students to view it without the proper safety precautions while getting to and from their buses. That's why some school districts are  canceling classes early so kids can enjoy the event safely with their families.

Dr. Avnish Deobhakta, vitreoretinal surgeon at New York Eye and Ear Infirmary at Mount Sinai, said parents should also be careful because it can be difficult for children to listen or keep solar eclipse glasses on. 

"You want to actually, in my opinion, kind of avoid them even looking at the eclipse, if possible," he said. "Never look directly at the sun, always wear the right eclipse sunglasses if you are going to look at the sun and make sure that those are coming from a reliable source."

Brinton recommends everyone starts their eclipse "viewing" early, by looking at professional photos and videos of an eclipse online or visiting a local planetarium. 

That way, you "have an idea of what to expect," he said. 

He also recommends the foundation  Prevent Blindness , which has resources for families about eclipse safety.

What happens if you look at a solar eclipse without eclipse glasses?

While your eyes likely won't hurt in the moment if you look at the eclipse without protection, due to lowered brightness and where damage occurs in the eye, beware: The rays can still cause damage .

The harm may not be apparent immediately. Sometimes trouble starts to appear one to a few days following the event. It could affect just one or both eyes.

And while some will regain normal visual function, sometimes the damage is permanent. 

"Often there will be some recovery of the vision in the first few months after it, but sometimes there is no recovery and sometimes there's a degree to which it is permanent," Brinton said. 

How long do you have to look at the eclipse to damage your eyes?

Any amount of time looking at the eclipse without protection is too long, experts say. 

"If someone briefly looks at the eclipse, if it's extremely brief, in some cases there won't be damage. But damage can happen even within a fraction of a second in some cases," Brinton said. He said he's had patients who have suffered from solar retinopathy, the official name for the condition.

Deobhakta treated a patient who watched the 2017 solar eclipse for 20 seconds without proper eye protection. She now has permanent damage in the shape of a crescent that interferes with her vision. 

"The crescent that is burned into the retina, the patient sees as black in her visual field," he said. "The visual deficit that she has will never go away."

How to know if you've damaged your eyes from looking at the eclipse

Signs and symptoms of eye damage following an eclipse viewing include headaches, blurred vision, dark spots, changes to how you see color, lines and shapes. 

Unfortunately, there isn't a treatment for solar retinopathy.

"Seeing an eye care professional to solidify the diagnosis and for education I think is reasonable," Brinton said, but added, "right now there is nothing that we do for this. Just wait and give it time and the body does tend to heal up a measure of it."

Sara Moniuszko is a health and lifestyle reporter at CBSNews.com. Previously, she wrote for USA Today, where she was selected to help launch the newspaper's wellness vertical. She now covers breaking and trending news for CBS News' HealthWatch.

More from CBS News

Couple gets engaged on flight to see total solar eclipse

Bill Nye shares tips for eclipse: "Be in the moment"

How to get a lower mortgage rate this spring

How to find the best tax relief company

Newsom called it a ‘gimmick.’ Now he’s using the trick to lower California’s massive deficit

• Show more sharing options
• Copy Link URL Copied!

With a windfall of cash five years ago, Gov. Newsom said he was doing away with a state budget “gimmick” one of his predecessors relied on to shave about $800 million off a deficit during the Great Recession.

The accounting trick, adopted in 2009, delayed state worker payroll from the end of one fiscal year on June 30 to the start of the next on July 1. A decade later Newsom spent nearly $1 billion to end the subterfuge, with one caveat.

“If I use it in six years, in a recession, forgive me,” Newsom said.

At his urging, Newsom and lawmakers have agreed to use the budget gimmick next year even though California isn’t in a recession.

The tactic is one of several maneuvers Democrats are relying on to reduce a historic budget deficit of at least $37.9 billion by pushing their spending problem forward to another year.

Of the $17.3 billion in budget reductions Newsom and Democrats have agreed to so far, only $3.6 billion are actual cuts.

Lawmakers made the first of those cuts Thursday and passed a budget trailer bill that lowers unspent funding allocations in 2022-23 and 2023-24 by $1.6 billion. Though Newsom touted the changes as part of an “early action” deal to shrink the deficit in Apri l, many of the reductions won’t be reflected in legislation until June or later.

At least so far, Newsom and lawmakers are largely relying on mechanisms other than cuts to shrink the deficit: borrowing $5.2 billion, delaying and deferring $5.2 billion in funding for state-sponsored programs to subsequent years and tapping into $3.4 billion from separate state funds. Democrats also agreed to draw down at least another $12.2 billion from the rainy day fund to cover their spending.

Budget watchers and Republican lawmakers criticized the strategy, saying that resorting to clever accounting now and dipping into California’s savings account while the economy remains strong will make the state more vulnerable to drastic cuts if a recession hits in the years ahead and revenue declines.

Newsom’s critics blame the governor and Democrats for overspending and causing the deficit. The “gimmick” is an example of what his critics view as Democrats failing to make the kind of tough choices that California households are forced to weigh when they spend more money than they bring in.

“They are doing things that you normally do in a recession and there is no recession here,” said David Crane, president of Govern for California, a nonprofit that seeks to oppose the influence of labor unions on state government. “You shouldn’t have to be dipping into reserves to meet a budget deficit if your revenues are 50% higher than they were when you came into office.”

Newsom and Democratic lawmakers detail first California budget cuts totaling $17 billion

Democratic lawmakers and Gov. Gavin Newsom are scrambling to lessen California’s budget deficit, which Newsom estimated at $37.9 billion in January.

April 4, 2024

General fund revenues, which the state uses to pay for most public services, were $140 billion when Newsom took office in in 2018-19. The governor’s January budget assumes revenues of more than $214 billion, an increase of 53%, for the upcoming fiscal year when Democrats plan to cut the rainy day fund in half.

According to the UCLA Anderson Forecast in March , California’s economy is growing faster than the rest of the nation and the possibility of a U.S. recession is fading. Newsom regularly trumpets the strength of the state economy.

“While there are still challenges ahead — notably, state and local government finance, homelessness and out-migration — the forces driving California’s economy remain robust,” UCLA economists wrote.

H.D. Palmer, a spokesperson for the governor’s Department of Finance, said the cuts Democrats have agreed to so far are part, but not all of the solution to the budget woes with more decisions coming in June. He also pointed to the fact that more than 70% of the general fund is spent on K-12 education, healthcare and human services.

“If you don’t agree with these solutions, that’s fine. What specific proposals would you offer up to offset that in terms of programmatic reductions?” he asked of budget critics.

Assembly Republican leader James Gallagher of Yuba City said he would start by funding the fundamentals, such as education, infrastructure and public safety, and then deciding what else the state has resources for.

Newsom often promotes all the one-time funding in his prior budgets, which he said would be easy to stop if the state swung from surplus to shortfall. But he has continued to support many of his expensive political priorities, such as the expansion of Medi-Cal to all eligible low-income immigrants, regardless of their legal status. A state audit also found that California has failed to monitor the effectiveness of its costly homelessness programs , which Newsom and lawmakers have spent $20 billion on over the last five years.

California spent billions on homelessness without tracking if it worked

The California State Auditor’s Office analyzed homelessness spending at the behest of lawmakers concerned about efficacy.

April 9, 2024

“A $73-billion deficit is no joke,” Gallagher said. “It’s a serious problem we’ve got to address. It makes me think that the governor just wants to ride this out until the end of his term and just leave this problem to somebody else.”

A combination of delayed tax deadlines and overspending based on inaccurate budget projections created the budget shortfall, which happens when spending exceeds projected revenues.

Newsom and lawmakers expected revenues to fall below projections because of a declining stock market, high interest rates and increased inflation, but the deficit is much worse than the state accounted for last June. The Newsom administration last pegged the deficit at $37.9 billion in January, though a more recent estimate from the Legislative Analyst’s Office suggests it could be up to $73 billion by the time the governor unveils his revised budget proposal in mid-May.

The state budget in California is largely dependent on income taxes paid by its highest earners. Revenues are prone to volatility, hinging on capital gains from investments, bonuses to executives and windfalls from new stock offerings, and notoriously difficult for the state to predict.

The governor repeatedly blames the shortfall problem on a decision by the federal government to delay the deadline to file 2022 income tax returns from April to November of last year due to winter storms.

In a typical budget year, state government has tax receipts in hand before the governor unveils a revised budget proposal in mid-May and before he reaches a final spending agreement with lawmakers in June. The tax delay forced lawmakers and the governor to enact the current budget in July based on estimates of how much money the state would collect in tax revenues by the November deadline. Those estimates were wildly off.

The legislation approved Thursday goes back and reduces unspent funding in the prior and current budget years. The changes include cutting $45 million for protecting communities from wildfire, $88 million for watershed resilience and reduces funding to expand broadband internet access by $34 million, among other trims.

The bill was part of the “early action” lawmakers and the governor announced that they would take in April to lower the deficit by $17.3 billion before the May revise. But only $3.3 billion of the reductions he claimed they would make can be adopted in law now, and the majority will be included in the final budget agreement, along with other reductions, approved this summer.

Newsom and Democrats announce a plan to reduce the enormous budget deficit. How? TBD

Facing a shortfall of at least $38 billion, Gov. Gavin Newsom and legislative leaders said they have agreed to $12 billion to $18 billion in cuts but offered no details.

March 21, 2024

“We put forward this early action plan to protect our progress and safeguard core programs so that we could spend time and energy on the more challenging decisions to close the remaining budget gap responsibly,” said Senate President Pro Tem Mike McGuire (D-Healdsburg) during floor debates in the Senate on Thursday. “And we’re gonna do just that.”

Democrats are attempting to offset the budget crisis before May, when an updated estimate might show an even deeper shortfall. Democrats also took an unusual step of requiring the state finance department to subtract the $17.3 billion from the estimated deficit before the budget is revised in May, making the shortfall appear smaller before many of the changes are reflected in law.

“This budget is nothing but smoke and mirrors, backroom deals done by the party in control,” said Sen. Brian Dahle (R-Bieber).

Delaying payroll from June 30, 2025 to July 1, 2025 is among the changes that Democrats agreed to, but will not vote on until this summer. The move is expected to push $1.6 billion in payroll costs to the 2025-2026 budget year.

While Crane, a political donor of Assembly Speaker Robert Rivas and dozens of other lawmakers, opposes Newsom’s decision to use the budget gimmick again, he said the “biggest sin” is the decision to dip into the state’s reserves in the absence of a recession. Newsom will have to declare a budget emergency in order to do so under state law.

“My one hope is that by the time the May revise comes around, he’s able to say I’m no longer going to have to dive into reserves,” Crane said.

More to Read

Column: Sore losers are trying to recall Newsom, again. They’re robbing voters of their voice

March 1, 2024

Voters say California’s budget deficit is an ‘extremely serious’ problem, poll finds

Jan. 18, 2024

Newsom cuts $2.9 billion from California climate programs, delays an additional $1.9 billion

Jan. 11, 2024

Start your day right

Sign up for Essential California for news, features and recommendations from the L.A. Times and beyond in your inbox six days a week.

You may occasionally receive promotional content from the Los Angeles Times.

Taryn Luna covers Gov. Gavin Newsom and California politics in Sacramento for the Los Angeles Times.

More From the Los Angeles Times

Dianne Feinstein made history. Now she could get honored with a post office?

April 11, 2024

World & Nation

Iowa governor signs bill that gives state authority to arrest and deport migrants

April 10, 2024

Impeached Homeland Security secretary navigates working with Republicans who want him out

Cornel West names L.A. professor, activist Melina Abdullah as running mate on presidential ticket

• Share full article

For more audio journalism and storytelling, download New York Times Audio , a new iOS app available for news subscribers.

How Tesla Planted the Seeds for Its Own Potential Downfall

Elon musk’s factory in china saved his company and made him ultrarich. now, it may backfire..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Katrin Bennhold. This is “The Daily.”

[MUSIC PLAYING]

Today, the story of how China gave Tesla a lifeline that saved the company — and how that lifeline has now given China the tools to beat Tesla at its own game. My colleague, Mara Hvistendahl, explains.

It’s Tuesday, April 9.

So, Mara, you’ve spent the past four months investigating Elon Musk and his ties to China through his company, Tesla. Tell us why.

Well, a lot of American companies are heavily invested in China, but Tesla’s kind of special. As my colleagues and I started talking to sources, we realized that many people felt that China played a crucial role in rescuing the company at a critical moment when it was on the brink of failure and that China helps account for Tesla’s success, for making it the most valuable car company in the world today, and for making Elon Musk ultra rich.

That’s super intriguing. So maybe take us back to the beginning. When does the story start?

So the story starts in the mid 2010s. Tesla had been this company that had all this hype around it. But —

A lot of people were shocked by Tesla’s earnings report. Not only did they make a lot less money than expected, they’re also making a lot less cars.

Tesla was struggling.

The delivery of the Model 3 has been delayed yet again.

Tesla engineers are saying 40 percent of the parts made at the Fremont factory need reworking.

At the time, they made their cars in Fremont, California, and they were facing production delays.

Tesla is confirming that Cal/OSHA is investigating the company over concerns over workplace safety.

Elon Musk has instituted a kind of famously grueling work culture at the factory, and that did not go over well with California labor law.

The federal government now has four active investigations involving Tesla.

They were clashing with regulators.

The National Transportation Safety Board will investigate a second crash involving Tesla’s autopilot system.

Billionaire entrepreneur Elon Musk — friends are really concerned about him. That’s what Musk told “The New York Times.”

And by 2018, he was having all of these crises.

According to “The Times,” Musk choked up multiple times and struggled to maintain his composure during an hour-long interview about turmoil at his electric car company, Tesla.

So all of this kind of converged to put immense pressure on him to do something.

And where does China come in?

Well, setting up a factory in China, in a way, would solve some of these problems for Musk. Labor costs were lower. Workers couldn’t unionize there. China provided access to this steady supply of cheaper parts. So Elon Musk was set on going to China. But first, Tesla and Musk wanted to change a key policy in China.

Hmm, what kind of policy?

So they wanted China to adopt a policy that was aimed at lowering car emissions. And the idea was that it would be modeled after a similar policy in California that had benefited Tesla there.

OK, so explain what that policy actually did. And how did it benefit Tesla?

So California had this system called the Zero-Emission Vehicle program. And that was designed to encourage companies to make cleaner cars, including electric vehicles. And they did that by setting pollution targets. So companies that made a lot of clean cars got credits. And then companies that failed to meet those targets, that produced too many gas-guzzling cars, would have to buy credits from the cleaner companies.

So California is trying to incentivize companies to make cleaner cars by forcing the traditional carmakers to pay cleaner car makers, which basically means dirtier car makers are effectively subsidizing cleaner cars.

Yes, that’s right. And Tesla, as a company that came along just making EVs, profited immensely from this system. And in its early years, when Tesla was really struggling to stay afloat, the money that it earned from selling credits in California to polluting car companies were absolutely crucial, so much so that the company structured a lot of its lobbying efforts around this system, around preserving these credits. And we talked to a former regulator who said as much.

How much money are we talking about here?

So from 2008, when Tesla unveiled its first car, up until the end of last year, Tesla made almost $4 billion by selling credits in California.

Wow. So Musk basically wants China to recreate this California-style program, which was incredibly lucrative for Tesla, there. And they’re basically holding that up as a condition to their building a factory in China.

Right. And at this point in the story, an interesting alliance emerges. Because it wasn’t just Tesla that wanted this emissions program in China. It was also environmentalists from California who had seen the success of the program up close in their own state.

If you go back to that period, to the early 2010s, I was living in China at the time in Beijing and Shanghai. And it was incredibly polluted. We called it airpocalypse at times. I had my first child in China at that point. And as soon as it was safe to put a baby mask on her, we put a little baby mask on her. There were days where people just would try to avoid going outside because it was so polluted. And some of the pollution was actually wafting across the Pacific Ocean to California.

Wow, so California is experiencing that Chinese air pollution firsthand and, in a way, has a direct stake in lowering it.

That’s right. So Governor Jerry Brown, for example — this became kind of his signature issue, was working with China to clean up the environment, in part by exporting this emission scheme. It was also an era of a lot more US-China cooperation. China was seen as absolutely crucial to combating climate change.

So you had all these groups working to get this California emissions scheme exported to China — and the governor’s office and environmental groups and Tesla. And it worked. In 2017, China did adopt a system that was modeled after California’s.

It’s pretty incredible. So California basically exports its emissions-trading system to China, which I imagine at the time was a big win for Californian environmentalists. But it was also a big win for Tesla.

It was definitely a big win for Tesla. And we know that in just a few years Tesla, made almost $1 billion from the emissions-trading program he helped lobby for in China.

So Elon Musk goes on, builds a factory in China. And he does so in Shanghai, where he builds a close relationship with the top official in the city, who actually is now the number-two official in all of China, Li Qiang.

So according to Chinese state media, Elon Musk actually proposed building the factory in two years, which would be fast. And Li came back and proposed that they do it in one year, which — things go up really quickly in China. But even for China, this is incredibly fast. And they broke ground on the factory in January 2019. And by the end of the year, cars were rolling off the line. So then in January 2020, Musk was able to get up on stage in Shanghai and unveil the first Chinese-made Teslas.

Really want to thank the Tesla team and the government officials that have been really helpful in making this happen.

Next to him on stage is Tesla’s top lobbyist who helped push through some of these changes.

Thank you. Yeah, everybody can tell Elon’s super, super happy today.

[SPEAKING CHINESE]

And she says —

Music, please.

Cue the music. [UPBEAT MUSIC]

And he actually broke into dance. He was so happy, a kind of awkward dance.

[UPBEAT MUSIC]

And what is the factory like?

The Shanghai factory is huge. 20,000 people work there. Tesla’s factories around the world tend to be pretty large, but the Shanghai workers work more shifts. And when Tesla set up in China, Chinese banks ended up offering Tesla $1.5 billion in low-interest loans. They got a preferential tax rate in Shanghai.

This deal was so generous that one auto industry official we talked to said that a government minister had actually lamented that they were giving Tesla too much. And it is an incredibly productive factory. It’s now the flagship export factory for Tesla.

So it opens in late 2019. And that’s, of course, the time when the pandemic hits.

Yes. I mean, you might think that this is really poor timing for Elon Musk. But it didn’t quite turn out that way. In fact, Tesla’s factory in Shanghai was closed for only around two weeks, whereas the factory in Fremont was closed for around two months.

That’s a big difference.

Yes, and it really, really mattered to Elon Musk. If you can think back to 2020, you might recall that he was railing against California politicians for closing his factory. In China, the factory stayed open. Workers were working around the clock. And Elon Musk said on a podcast —

China rocks, in my opinion.

— China rocks.

There’s a lot of smart, hardworking people. And they’re not entitled. They’re not complacent, whereas I see —

We’ve seen a lot of momentum and enthusiasm for electric vehicles, stocks, and Tesla certainly leading the charge.

Tesla’s stock price kept going up.

Tesla has become just the fifth company to reach a trillion-dollar valuation. The massive valuation happened after Tesla’s stock price hit an all-time high of more than $1,000.

So this company that had just a few years earlier been on the brink of failure, looking to China for a lifeline, was suddenly riding high. And —

Tesla is now the most valuable car company in the world. It’s worth more than General Motors, Ford, Fiat, Chrysler.

By the summer, it had become the most valuable car company in the world.

Guess what? Elon Musk is now the world’s richest man.

“Forbes” says he’s worth more than $255 billion.

And Elon Musk’s wealth is tied up in Tesla stock. And in the following year, he became the wealthiest man in the world.

So you have this emission trading system, which we discussed and which, in part, thanks to Tesla, is now established in China. It’s bringing in money to Tesla. And now this Shanghai factory is continuing to produce cars for Tesla in the middle of the pandemic. So China really paid off for Tesla. But what was in it for China?

Well, China wasn’t doing this for charity.

What Chinese leaders really wanted was to turn their fledgling electric vehicle industry into a global powerhouse. And they figured that Tesla was the ticket to get there. And that’s precisely what happened.

We’ll be right back.

So, Mara, you’ve just told us the story of how Elon Musk used China to turn Tesla into the biggest car maker in the world and himself — at one point — into the richest man in the world. Now I want to understand the other side of this story. How did China use Tesla?

Well, Tesla basically became a catfish for China’s EV industry.

A catfish, what do you mean by that?

It’s a term from the business world. And, essentially, it means a super aggressive fish that makes the other fish in the pond swim faster. And by bringing in this super competitive, aggressive foreign company into China, which at that point had these fledgling EV companies, Chinese leaders hoped to spur the upstart Chinese EV makers to up their game.

So you’re saying that at this point, China actually already had a number of smaller EV companies, which many people in the West may not even be aware of, these smaller fish in the pond that you were referring to.

Yes, there were a lot of them. They were often locally based. Like, one would be strong in one city, and one would be strong in another city. And Chinese leaders saw that they needed to become more competitive in order to thrive.

And China had tried for decades to build up this traditional car industry by bringing in foreign companies to set up joint ventures. They had really had their sights set on building a strong car industry, and it didn’t really work. I mean, how many traditional Chinese car company brands can you name?

Exactly none.

Yeah, right. So going back to the aughts and the 2010s, they had this advantage that many Chinese hadn’t yet been hooked on gas-guzzling cars. There were still many people who were buying their first car ever. So officials had all these levers they could pull to try to encourage or try to push people’s behavior in a certain direction.

And their idea was to try to ensure that when people went to buy their first car, it would be an EV — and not just an EV but, hopefully, a Chinese EV. So they did things like — at the time, just a license plate for your car could cost an exorbitant amount of money and be difficult to get. And so they made license plates for electric vehicles free. So there were all these preferential policies that were unveiled to nudge people toward buying EVs.

So that’s fascinating. So China is incentivizing consumers to buy EV cars and incentivizing also the whole industry to get its act together by chucking this big American company in the mix and hoping that it will increase competitiveness. What I’m particularly struck by, Mara, in what you said is the concept of leapfrogging over the conventional combustion engine phase, which took us decades to live through. We’re still living in it, in many ways, in the West.

But listening to you, it sounds a little bit like China wasn’t really thinking about this transition to EVs as an environmental policy. It sounds like they were doing this more from an industrial-policy perspective.

Right. The environment and the horrible era at the time was a factor, but it was a pretty minor factor, according to people who were privy to the policy discussions. The more significant factor was industrial policy and an interest in building up a competitive sphere.

So China now wants to become a leader in the global EV sector, and it wants to use Tesla to get there. What does that actually look like?

Well, you need sophisticated suppliers to make the component parts of electric vehicles. And just by being in China, Tesla helped spur the development of several suppliers. Like, for example, the battery is a crucial piece of any EV.

And Tesla, with a fair amount of encouragement — and also various levers from the Chinese government — became a customer of a battery maker called CATL, a homegrown Chinese battery maker. And they have become very close to Tesla and have even set up a factory near Teslas in Shanghai. And today, with Tesla’s business — and, of course, with the business of some other companies — CATL is the biggest battery maker in the world.

But beyond just stimulating the growth of suppliers, Tesla also made these other fish in the pond swim faster. And the biggest Chinese EV company to come out of that period is one called BYD. It’s short for Build Your Dreams.

We are BYD. You’ve probably never heard of us.

From battery maker to the biggest electric vehicle or EV manufacturer in China.

They’ve got a lot of models. They’ve got a lot of discounts. They’ve got a lot of market growth.

China’s biggest EV maker just overtook Tesla in terms of worldwide sales.

BYD 10, Chinese automobile redefined.

I’ve actually started seeing that brand on the streets here in Europe recently, especially in Germany, where my brother actually used to lease a Tesla and now leases a BYD.

Does he like it?

He does. Although he did, to be fair, say that he misses the luxury of the Tesla, but it just became too expensive, really.

The price point is a huge reason that BYD is increasingly giving Tesla a run for its money. Years ago, back in 2011 —

Although there’s competitors now ramping up. And, as you’re familiar with, BYD, which is also —

— Elon Musk actually mocked their cars.

— electric vehicles, here he is trying to compete. Why do you laugh?

He asked an interviewer —

Have you seen their car?

I have seen their car, yes.

— have you seen their cars? Sort of suggesting, like, they’re no competition for us.

You don’t see them at all as a competitor?

Why is that? I mean, they offer a lower price point.

I don’t think they have a great product. I think their focus is — and rightly should be — on making sure they don’t die in China.

But they have been steadily improving. They’ve been in the EV space for a while, but they really started improving a few years ago, once Tesla came on the scene. That was due to a number of factors, not entirely because of Tesla. But Tesla played a role in helping train up talent in China. One former Tesla employee who worked at the company as they were getting set up in China told me that most of the employees who were at the company at the time now work for Chinese competitors.

So they have really played this important role in the EV ecosystem.

And you mentioned the price advantage. So just for comparison, what does an average BYD sell for compared to a more affordable Tesla car?

So BYD has an ultra-cheap model called the Seagull that sells for around $10,000 now in China, whereas Tesla Model 3s and Model Ys in China sell for more than twice that.

Wow. How’s BYD able to sell EVs at these much lower prices?

Well, the Seagull is really just a simpler car. It has less range than a Tesla. It lacks some safety measures. But BYD has this other crucial advantage, which is that they’re vertically integrated. Like, they control many aspects of the supply chain, up and down the supply chain. When you look at the battery level, they make batteries. But they even own the mines where lithium is mined for the batteries.

And they recently launched a fleet of ships. So they actually operate the boats that are sending their cars to Europe or other parts of the world.

So BYD is basically cutting out the middleman on all these aspects of the supply chain, and that’s how they can undercut other car makers on price.

Yeah. They’ve cut out the middleman, and they’ve cut out the shipping company and almost everything else.

So how is BYD doing now as a company compared to Tesla?

In terms of market cap, they’re still much smaller than Tesla. But, crucially, they overtook Tesla in sales in the last quarter of last year.

Yeah, that was a huge milestone. Tesla still dominates in the European market, which is a very important market for EVs. But BYD is starting to export there. And Europe traditionally is kind of automotive powerhouse, and the companies and government officials there are very, very concerned. I interviewed the French finance minister, and he told me that China has a five - to seven-year head start on Europe when it comes to EVs.

Wow. And what has Elon Musk said about this incredible rise of BYD in recent years? Do you think he anticipated that Tesla’s entry into the Chinese market could end up building up its own competition?

Well, I can’t get inside his head, and he did not respond to our questions. But —

The Chinese car companies are the most competitive car companies in the world.

— he has certainly changed his tune. So, remember, he was joking about BYD some years ago.

Yeah, he’s not joking anymore.

I think they will have significant success.

He had dismissed Chinese EV makers. He now appears increasingly concerned about these new competitors —

Frankly, I think if there are not trade barriers established, they will pretty much demolish most other car companies in the world.

— to the point that on an earnings call in January, he all but endorsed the use of trade barriers against them.

They’re extremely good.

I think it’s so interesting, in a way — of course, with perfect hindsight — the kind of maybe complacency or naivete with which he may not have anticipated this turn of events. And in some ways, he’s not alone, right? It speaks to something larger. Like, China, for a long time, was seen as kind of the sweatshop or the manufacturer of the world — or perhaps as an export market for a lot of these Western companies. It certainly wasn’t putting out its own big brand names. It was making stuff for the brand names.

But recently, they have quite a lot of their own brand names. Everybody talks about TikTok. There’s Huawei. There’s WeChat, Lenovo. And now there is BYD. So China is becoming a leader in technology in certain areas. And I think that shift in some ways has happened. And a lot of Western companies — perhaps like Tesla — were kind of late to waking up to that.

Right. Tesla is looking fragile now. Their stock price dropped 30 percent in the first quarter of this year. And to a large degree, that is because of the threat of companies like BYD from China and the perception that Tesla’s position as number one in the market is no longer guaranteed.

So, Mara, all this raises a much bigger question for me, which is, who is going to own the future of EVs? And based on everything you’ve said so far, it seems like China owns the future of EVs. Is that right?

Well, possibly, but the jury is still out. Tesla is still far bigger for now. But there is this increasing fear that China owns the future of EVs. If you look at the US, there are already 25 percent tariffs on EVs from China. There’s talk of increasing them. The Commerce Department recently launched an investigation into data collection by electric vehicles from China.

So all of these factors are creating uncertainty around what could happen. And the European Union may also add new tariffs against Chinese-made cars. And China is an economic rival and a security rival and, in many ways, our main adversary. So this whole issue is intertwined with national security. And Tesla is really in the middle of it.

Right. So the sort of new Cold War that people are talking about between the US and China is, in a sense, the backdrop to this story. But on one level, what we’ve been talking about, it’s really a corporate story, an economic story that has this geopolitical backdrop. But it’s also very much an environmental story. So, regardless of how Elon Musk and Tesla fare in the end, is BYD’s rise and its ability to create high-quality and — perhaps more importantly — affordable EVs ultimately a good thing for the world?

If I think back on those years I spent living in Shanghai and Beijing when it was extremely polluted and there were days when you couldn’t go outside — I don’t think anyone wants to go back to that.

So it’s clear that EVs are the future and that they’re crucial to the green energy transition that we have to make. How exactly we get there is still unclear. But what is true is that China did just make that transition easier.

Mara, thank you so much.

Thank you, Katrin.

Here’s what else you need to know today.

[CROWD CHEERING]

Millions of people across North America were waiting for their turn to experience a rare event on Monday. From Mexico —

Cuatro, tres, dos, uno.

— to Texas.

Awesome, just awesome.

We can see the corona really well. Oh, you can see —

[BACKGROUND CHATTER]

Oh, and we are falling into darkness right now. What an incredible sensation. And you are hearing and seeing the crowd of 15,000 gathered here in south Illinois.

Including “Daily” producers in New York.

It’s like the sky is almost —

— like a deep blue under the clouds.

Wait, look. It’s just —

Oh my god. The sun is disappearing. And it’s gone. Oh. Whoa.

All the way up to Canada.

Yeah, that’s what I’m talking about. That’s what I’m talking about.

The moon glided in front of the sun and obscured it entirely in a total solar eclipse, momentarily plunging the day into darkness.

It’s super exciting. It’s so amazing to see science in action like this.

Today’s episode was produced by Rikki Novetsky and Mooj Zadie with help from Rachelle Bonja. It was edited by Lisa Chow with help from Alexandra Leigh Young, fact checked by Susan Lee, contains original music by Marion Lozano, Diane Wong, Elisheba Ittoop, and Sophia Lanman and was engineered by Chris Wood.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m catching Katrin Bennhold. See you tomorrow.

• April 11, 2024   •   28:39 The Staggering Success of Trump’s Trial Delay Tactics
• April 10, 2024   •   22:49 Trump’s Abortion Dilemma
• April 9, 2024   •   30:48 How Tesla Planted the Seeds for Its Own Potential Downfall
• April 8, 2024   •   30:28 The Eclipse Chaser
• April 7, 2024 The Sunday Read: ‘What Deathbed Visions Teach Us About Living’
• April 5, 2024   •   29:11 An Engineering Experiment to Cool the Earth
• April 4, 2024   •   32:37 Israel’s Deadly Airstrike on the World Central Kitchen
• April 3, 2024   •   27:42 The Accidental Tax Cutter in Chief
• April 2, 2024   •   29:32 Kids Are Missing School at an Alarming Rate
• April 1, 2024   •   36:14 Ronna McDaniel, TV News and the Trump Problem
• March 29, 2024   •   48:42 Hamas Took Her, and Still Has Her Husband
• March 28, 2024   •   33:40 The Newest Tech Start-Up Billionaire? Donald Trump.

Hosted by Katrin Bennhold

Featuring Mara Hvistendahl

Produced by Rikki Novetsky and Mooj Zadie

With Rachelle Bonja

Edited by Lisa Chow and Alexandra Leigh Young

Original music by Marion Lozano ,  Diane Wong ,  Elisheba Ittoop and Sophia Lanman

Engineered by Chris Wood

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music

When Elon Musk set up Tesla’s factory in China, he made a bet that brought him cheap parts and capable workers — a bet that made him ultrarich and saved his company.

Mara Hvistendahl, an investigative reporter for The Times, explains why, now, that lifeline may have given China the tools to beat Tesla at its own game.

On today’s episode

Mara Hvistendahl , an investigative reporter for The New York Times.

Background reading

A pivot to China saved Elon Musk. It also bound him to Beijing .

Mr. Musk helped create the Chinese electric vehicle industry. But he is now facing challenges there as well as scrutiny in the West over his reliance on China.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

Fact-checking by Susan Lee .

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Katrin Bennhold is the Berlin bureau chief. A former Nieman fellow at Harvard University, she previously reported from London and Paris, covering a range of topics from the rise of populism to gender. More about Katrin Bennhold

Mara Hvistendahl is an investigative reporter for The Times focused on Asia. More about Mara Hvistendahl

Advertisement