SAT / ACT Prep Online Guides and Tips

How to Find the Mean of a Set of Numbers: Formula and Examples

Posted by Hannah Muniz | May 20, 2019 5:00:00 PM

General Education

 

feature_math_board

Are you taking the SAT or ACT and want to make sure you know how to work with data sets? Or maybe you’re looking to refresh your memory for a high school or college math class. Whatever the case, it’s important you know how to find the mean of a data set.

We'll explain what the mean is used for in math, how to calculate the mean, and what problems about the mean can look like.

 

What Is a Mean and What Is It Used For?

The mean, or arithmetic mean, is the average value of a set of numbers. More specifically, it's the measure of a "central" or typical tendency in a given set of data.

Meanoften simply called the "average"—is a term used in statistics and data analysis. In addition, it's not unusual to hear the words "mean" or "average" used with the terms "mode," "median," and "range," which are other methods of calculating the patterns and common values in data sets.

Briefly, here are the definitions of these terms:

  • Mode  the value that appears most frequently in a data set
  • Median  the middle value of a data set (when arranged from lowest value to highest)
  • Range  the difference between the highest and smallest values in a data set

So what is the purpose of the mean exactly? If you have a data set with a wide range of numbers, knowing the mean can give you a general sense of how these numbers could essentially be put together into a single representative value.

For example, if you’re a high school student getting ready to take the SAT, you might be interested to know the current mean SAT score. Knowing the mean score gives you a rough idea of how most students taking the SAT tend to score on it.

 

How to Find the Mean: Overview

To find the arithmetic mean of a data set, all you need to do is add up all the numbers in the data set and then divide the sum by the total number of values.

Let’s look at an example. Say you’re given the following set of data:

$$6, 10, 3, 27, 19, 2, 5, 14$$

To find the mean, you’ll first need to add up all the values in the data set like this:

$$6 + 10 + 3 + 27 + 19 + 2 + 5 + 14$$

Note that you don’t need to rearrange the values here (though you may if you wish to) and can simply add them in the order in which they’ve been presented to you.

Next, write down the sum of all the values:

$$6 + 10 + 3 + 27 + 19 + 2 + 5 + 14 = \bo86$$

The last step is to take this sum (86) and divide it by the number of values in the data set. Because there are eight different values (6, 10, 3, 27, 19, 2, 5, 14), we'll be dividing 86 by 8:

$$86 / 8 = 10.75$$

The mean, or average, for this set of data is 10.75.

 

body_calculator_pen

 

How to Calculate a Mean: Practice Questions

Now that you know how to find the average—in other words, how to calculate the mean of a given set of data—it’s time to test what you’ve learned. In this section, we'll give you four math questions that involve finding or using the mean.

The first two questions are our own, whereas the second two are official SAT/ACT questions; as such, these two will require a little bit more thought.

Scroll past the questions for the answers and answer explanations.

 

Practice Question 1

Find the mean of the following set of numbers: 5, 26, 9, 14, 49, 31, 109, 5.  

 

Practice Question 2

You are given the following list of numbers: 4, 4, 2, 11, 6, $X$, 1, 3, 2. The arithmetic mean is 4. What is the value of $X$?

 

Practice Question 3

The list of numbers 41, 35, 30, $X, Y$, 15 has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the mean of the list?

  1. 20
  2. 25
  3. 26
  4. 27
  5. 30

Source: 2018-19 Official ACT Practice Test

 

Practice Question 4

At a primate reserve, the mean age of all the male primates is 15 years, and the mean age of all female primates is 19 years. Which of the following must be true about the mean age $m$ of the combined group of male and female primates at the primate reserve?

  1. $m = 17$
  2. $m > 17$
  3. $m < 17$
  4. $15 < m < 19$

Source: The College Board

 

body_solutions-1

 

How to Find the Average: Answers + Explanations

Once you’ve tried out the four practice questions above, it’s time to compare your answers and see whether you understand not just how to find the mean of data but also how to use what you know about the mean to more effectively approach any math questions that deal with averages.

Here are the answers to the four practice questions above:

  • Practice Question 1: 31
  • Practice Question 2: 3
  • Practice Question 3: C. 26
  • Practice Question 4: D. $15 < m < 19$

Keep reading to see the answer explanation for each question.

 

Practice Question 1 Answer Explanation

Find the mean of the following set of numbers: 5, 26, 9, 14, 49, 31, 109, 5.

This is a straightforward question that simply asks you to calculate the arithmetic mean of a given data set.

First, add up all the numbers in the data set (remember that you don’t need to arrange them in order from lowest to highestonly do this if you’re trying to find the median):

$$5 + 26 + 9 + 14 + 49 + 31 + 109 + 5 = \bo248$$

Next, take this sum and divide it by the number of values in the data set. Here, there are eight total values, so we'll divide 248 by 8:

$$248 / 8 = 31$$

The mean and correct answer is 31.

 

Practice Question 2 Answer Explanation

You are given the following list of numbers: 4, 4, 2, 11, 6, $X$, 1, 3, 2. The arithmetic mean is 4. What is the value of $X$?

For this question, you’re essentially working backward: you already know the mean and now must use this knowledge to help you solve for the missing value, $X$, in the data set.

Recall that to find the mean, you add up all the numbers in a set and then divide the sum by the total number of values.

Since we know the mean is 4, we’ll start by multiplying 4 by the number of values (there are nine separate numbers here, including $X$):

$$4 * 9 = 36$$

This gives us the sum of the data set (36). Now, the question becomes an algebra problem, in which all we need to do is simplify and solve for $X$:

$$4 + 4 + 2 + 11 + 6 + X + 1 + 3 + 2 = 36$$

$$33 + X = 36$$

$$X = 3$$

The correct answer is 3.

 

body_math_practicePractice makes perfect!

 

Practice Question 3 Answer Explanation

The list of numbers 41, 35, 30, $X, Y$, 15 has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the mean of the list?

  1. 20
  2. 25
  3. 26
  4. 27
  5. 30

This tricky-looking math problem comes from an official ACT practice test, so you can expect it to be a little less direct than your typical arithmetic mean problem.

Here, we’re given a data set with two unknown values:

 41, 35, 30, $X, Y$, 15

We’re also given two critical pieces of information:

  • The mode is 15
  • The median is 25

To solve for the mean of this data set, we will need to use all the information we’ve been given and will also need to know what the mode and median are.

As a reminder, the mode is the value that appears most frequently in a data set, while the median is the middle value in a data set (when all values have been arranged from lowest to highest).

Since the mode is 15, this must mean that the value 15 appears at least twice in the data set (in other words, more times than any other value appears). As a result, we can say replace either $X$ or $Y$ with 15:

$$41, 35, 30, X, 15, 15$$

We’re also told that the median is 25. To find the median, you must first rearrange the data set in order from lowest value to highest value.

Since the median is more than 15 but less than 30, we should put $\bi X$ between these two values. Here’s what we get when we rearrange our values from lowest to highest:

$$15, 15, X, 30, 35, 41$$

There are six values in total, (including $X$) meaning that the median will be the number exactly halfway between the third and fourth values in the data set. In short, 25 (the median) must come halfway between $X$ and 30.

This means that $X$ must equal 20, since that would put it 5 away from 20 and 5 away from 30 (or halfway between the two values).

We now have a complete data set with no unknown values:

$$15, 15, 20, 30, 35, 41$$

All we have to do now is use these values to solve for the mean. Start by adding them all up:

$$15 + 15 + 20 + 30 + 35 + 41 = 156$$

Finally, divide the sum by the number of values in the data set (that’s six):

$$156 / 6 = 26$$

The correct answer is C. 26.

 

Practice Question 4 Answer Explanation

At a primate reserve, the mean age of all the male primates is 15 years, and the mean age of all female primates is 19 years. Which of the following must be true about the mean age $m$ of the combined group of male and female primates at the primate reserve?

  1. $m = 17$
  2. $m > 17$
  3. $m < 17$
  4. $15 < m < 19$

This practice problem is an official SAT Math practice question from the College Board website.

For this math question, you’re not expected to solve for the mean but must instead use what you know about two means to explain what the mean of the larger group could be. Specifically, we're being asked how we can use these two means to express, in algebraic terms, the mean age ($\bi m$) for both male and female primates.

Here’s what we know: first, the mean age of all male primates is 15 years. Secondly, the mean age of all female primates is 19 years. This means that, in general, the female primates are older than the male primates.

Since the mean age for male primates (15) is lower than that for female primates (19), we know that the mean age for both groups cannot logically exceed 19 years.

Similarly, because the mean age for female primates is greater than that for male primates, we know that the mean age for both cannot logically fall below 15 years.

We are therefore left with the understanding that the mean age for the male and female primates together must be greater than 15 years (the mean age of the males) but also less than 19 years (the mean age of the females).

This rationale can be written as the following inequality:

$$15 < m < 19$$

The correct answer is D. 15 < $\bi m$ < 19.

 

What’s Next?

To learn even more about data sets, look at our guide to the best strategies for mean, median, and mode on SAT Math.

Taking the SAT or ACT soon? Then you'll definitely want to know what kind of math you're going to be tested on. Check out our in-depth guides to the SAT Math section and the ACT Math section to get started.

What are the most important math formulas to know for the SAT and ACT? Get an overview of the 28 critical SAT formulas and the 31 critical ACT formulas you should know.

 

Have friends who also need help with test prep? Share this article!
Hannah Muniz
About the Author

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.



Get Free Guides to Boost Your SAT/ACT
100% Privacy. No spam ever.

Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!