How To Calculate The Mean Of A Large Data Set

feature_math_board

Are you lot 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 example, information technology's important yous 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 hateful, and what problems well-nigh the hateful tin can expect like.

What Is a Mean and What Is Information technology Used For?

The mean, or arithmetic mean, is the boilerplate value of a set of numbers. More than specifically, it'due south the measure of a "central" or typical tendency in a given prepare of data.

Mean— often simply chosen the "average"—is a term used in statistics and data assay. 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 computing the patterns and common values in information 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 everyman 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 information set with a broad range of numbers, knowing the mean can give you a general sense of how these numbers could substantially be put together into a single representative value.

For instance, if you lot're a high school educatee getting ready to take the SAT, you might be interested to know the electric current mean SAT score. Knowing the hateful score gives you a rough idea of how most students taking the Sabbatum tend to score on it.

How to Find the Mean: Overview

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

Allow's look at an example. Say you lot're given the post-obit set of data:

$$6, 10, three, 27, 19, two, 5, 14$$

To observe the hateful, you'll outset need to add up all the values in the data set like this:

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

Note that you don't need to rearrange the values here (though you may if you wish to) and tin can simply add together them in the social club in which they've been presented to you lot.

Side by side, write down the sum of all the values:

$$6 + ten + 3 + 27 + 19 + 2 + v + fourteen = \bo86$$

The last step is to take this sum (86) and split up it by the number of values in the data set. Because at that place are eight different values (vi, 10, three, 27, 19, 2, 5, 14), we'll be dividing 86 by 8:

$$86 / 8 = 10.75$$

The mean, or boilerplate, for this ready of data is x.75.

body_calculator_pen

How to Calculate a Mean: Practice Questions

Now that you know how to notice the average—in other words,how to calculate the mean of a given set up of data—it's time to examination what you lot've learned. In this section, we'll requite you lot 4 math questions that involve finding or using the hateful.

The start two questions are our own, whereas the 2nd two are official SAT/Act questions; equally such, these two will crave a fiddling chip more idea.

Gyre past the questions for the answers and reply explanations.

Do Question 1

Discover the mean of the post-obit set of numbers: 5, 26, 9, xiv, 49, 31, 109, 5.

Practice Question 2

You are given the post-obit list of numbers: 4, four, ii, eleven, 6, $Ten$, ane, 3, 2. The arithmetic mean is iv. 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?

Source: 2022-19 Official Act Exercise Test

Practice Question 4

At a primate reserve, the mean age of all the male person 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 historic period $m$ of the combined group of male and female primates at the primate reserve?

$m = 17$
$m > 17$
$m < 17$
$fifteen < m < 19$

Source: The Higher Board

body_solutions-1

How to Find the Average: Answers + Explanations

Once you've tried out the four exercise questions above, it's fourth dimension to compare your answers and see whether you understand non just how to discover the mean of data merely besides how to use what you know about the mean to more than effectively approach whatever math questions that deal with averages.

Hither are the answers to the four exercise questions higher up:

Practice Question 1: 31
Practice Question ii: three
Exercise Question 3: C. 26
Practise Question four: D. $15 < m < 19$

Go on reading to meet the answer caption for each question.

Do Question 1 Answer Caption

Find the mean of the post-obit prepare of numbers: 5, 26, 9, 14, 49, 31, 109, 5.

This is a straightforward question that simply asks you to summate the arithmetics hateful of a given data gear up.

Showtime, add together up all the numbers in the data set (remember that you don't need to suit them in society from lowest to highest—only do this if yous're trying to find the median):

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

Next, have this sum and dissever information technology by the number of values in the data set. Hither, there are eight full values, so nosotros'll divide 248 past 8:

$$248 / 8 = 31$$

The mean and right respond is 31.

Practice Question ii Answer Caption

You are given the following list of numbers: 4, four, 2, 11, 6, $10$, i, 3, 2. The arithmetic mean is 4. What is the value of $X$?

For this question, you lot're essentially working backward: you already know the hateful and now must use this cognition to assist yous solve for the missing value, $10$, in the data gear up.

Retrieve that to find the mean, you lot add together up all the numbers in a gear up and then split up the sum by the total number of values.

Since we know the mean is 4, we'll start by multiplying iv by the number of values (in that location are nine carve up numbers hither, including $X$):

$$4 * 9 = 36$$

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

$$iv + 4 + two + 11 + half-dozen + 10 + 1 + iii + ii = 36$$

$$33 + 10 = 36$$

$$10 = 3$$

The correct answer is iii.

body_math_practice Practise makes perfect!

Practice Question three Answer Caption

The list of numbers 41, 35, 30, $10$, $Y$, fifteen has a median of 25. The mode of the list of numbers is 15. To the nearest whole number, what is the hateful of the listing?

This catchy-looking math problem comes from an official ACT practice examination, so you can expect it to be a picayune less straight than your typical arithmetic mean problem.

Here, nosotros're given a data set with two unknown values:

41, 35, 30, $X$, $Y$, xv

We're too given two disquisitional pieces of information:

The style 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 volition also need to know what the mode and median are.

Every bit a reminder, the style is the value that appears about often in a data set, while the median is the middle value in a information set (when all values have been arranged from lowest to highest).

Since the way is fifteen, this must mean that the value 15 appears at to the lowest degree twice in the data set (in other words, more than times than whatsoever other value appears). Every bit a result, we can say supersede either $Ten$ or $Y$ with 15:

41, 35, xxx, $X$,15,15

Nosotros're too 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 than 15 merely less than thirty, we should put $X$ between these 2 values. Here'due south what we get when we rearrange our values from lowest to highest:

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

There are six values in full, (including $X$) meaning that the median volition exist the number exactly halfway between the third and fourth values in the data set up.In brusk,25 (the median) must come halfway between $Ten$ and thirty.

This means that $X$ must equal 20, since that would put it 5 away from xx and 5 abroad from xxx (or halfway between the 2 values).

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

xv,15, 20, 30, 35, 41

All we take to do at present is use these values to solve for the mean. Start past calculation them all upward:

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 Respond Explanation

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

$one thousand = 17$
$m > 17$
$m < 17$
$15 < chiliad < 19$

This practice trouble is an official Saturday Math exercise question from the College Lath website.

For this math question, you're not expected to solve for the mean merely must instead use what you know about two means to explain what the hateful of the larger group could be. Specifically, we're being asked how we tin can utilize these two ways to express, in algebraic terms, the mean historic period ($\bi chiliad$) forboth male and female person primates.

Hither's what nosotros know: first, the mean age of all male primates is 15 years. Secondly, the mean age of all female primates is xix years. This ways that, in general, the female primates are older than the male primates.

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

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

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

This rationale can exist written as the following inequality:

$$15 < chiliad < nineteen$$

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

What's Next?

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

Taking the Sat or ACT soon? Then y'all'll definitely want to know what kind of math you're going to be tested on. Bank check out our in-depth guides to the Sat Math section and the Deed Math section to go started.

What are the most important math formulas to know for the Sat and ACT?Become an overview of the 28 disquisitional Sat formulas and the 31 critical Deed formulas you should know.

Demand more than assistance with this topic? Check out Tutorbase!

Our vetted tutor database includes a range of experienced educators who can help y'all smoothen an essay for English language or explain how derivatives piece of work for Calculus. You tin can apply dozens of filters and search criteria to find the perfect person for your needs.

Connect With a Tutor Now

Have friends who also need help with test prep? Share this commodity!

About the Author

Hannah received her MA in Japanese Studies from the Academy of Michigan and holds a bachelor's degree from the Academy of Southern California. From 2022 to 2022, she taught English in Japan via the JET Program. She is passionate most education, writing, and travel.

Source: https://blog.prepscholar.com/how-to-find-the-mean#:~:text=To%20find%20the%20arithmetic%20mean,the%20total%20number%20of%20values.