How To Find The Average In Math

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How to find the average in math

Here you will learn about how to find the average in math, including what the average is and how to find the average in math.

Students will first learn about how to find the average in math as part of statistics and probability in $6$ th grade and will expand on their knowledge throughout middle school and high school.

What is the average in math?

The average in math is calculated by finding the total, or sum, of the values and dividing the total by the number of values. This is also known as the arithmetic mean.

\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}

For example,

Find the average of the given numbers: $4, 12, 13, 15$ and $16.$

\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}=\cfrac{4+12+13+15+16}{5}=\cfrac{60}{5}=12

$12$ is the average of the data set.

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Types of averages

There are several types of averages that are used in mathematics, each summarizing data differently.

When asked to find the average in math, it is most common that you are being asked to find what is known as the arithmetic mean. It is important to note that the term average could actually refer to any measure of central tendency.

Median and mode are also measures of central tendency. While all three measure center in some way, they are not the same. Mean can be thought of as sharing equally between all data points.

For the purpose of this webpage, we will be focusing on what is most commonly referred to as ‘finding the average’, finding the total, or sum, of the values and dividing the total by the number of values.

What is the average in math?

$What is the average in math?$

Common Core State Standards

How does this relate to $6$ th grade math?

Grade 6 – Statistics and Probability (6.SP.A.3)
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

In order to find the average in math:

Find the sum of the data points.
Divide the sum by the number of data points.
Write down the answer.

How to find the average in math examples

Example 1: finding the average

Calculate the average of this list of numbers:

Find the sum of the data points.

5+7+11+12+15=50

2Divide the sum by the number of data points.

There are $5$ values in the data set. Divide the total by $5.$

\text { average }=\cfrac{\text { sum of the values }}{\text { number of values }}=\cfrac{50}{5}=10

3Write down the answer.

The average is $10.$

Example 2: finding the average

Calculate the average of this set of numbers to the nearest tenth.

Find the sum of the data points.

9+13+15+15+16+23=91

Divide the sum by the number of data points.

There are $6$ values in the data set. Divide the total by $6.$

$\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}=\cfrac{91}{6}=15.1666667$

Write down the answer.

$15.1666667$ to the nearest tenth is $15.2.$

$15.2$ is the average.

Example 3: finding the average

Calculate the average of this set of data to the nearest hundredth.

Find the sum of the data points.

12+13+15+16+20+22+24=122

Divide the sum by the number of data points.

There are $7$ values in the data set. Divide the total by $7.$

$\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}=\cfrac{122}{7}=17.4285 \ldots$

Write down the answer.

$17.4285 \ldots$ rounded to the nearest hundredth is $17.43.$

$17.43$ is the average.

Example 4: finding the mean

Calculate the average of this list of numbers:

Find the sum of the data points.

100+105+108+110+115=538

Divide the sum by the number of data points.

There are $5$ values in the data set. Divide the total by $5.$

$\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}=\cfrac{538}{5}=107.6$

Write down the answer.

$107.6$ is the average.

How to solve a problem involving the average in math

In order to solve a problem involving the average in math:

Use the average and number of values to find the total.
Find the sum of the known data points.
Subtract the sum of the known data points from the first total to find the missing data point.

Problem solving involving average in math examples

Example 5: problem solving

The average of $4$ values is $12.$

Here are $3$ of the values:

Find the $4^{th}$ value.

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

The average of $4$ values is $12.$ Multiply these together to find the total of the $4$ numbers.

$\text{Total of 4 values}=\text{average} \times \text{number of values}=12\times 4=48$

Find the sum of the known data points.

\text { Total of } 3 \text { values }=5+7+10=22

Subtract the sum of the known data points from the first total to find the missing data point.

48-22=26

The $4^{th}$ value is $26.$

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

$\begin{aligned} & \text { average }=\cfrac{\text { sum of values }}{\text { number of values }} \\\\ & 12=\cfrac{5+7+10+x}{4} \\\\ & 12=\cfrac{22+x}{4} \\\\ & 48=22+x \\\\ & 26=x \end{aligned}$

Example 6: problem solving

The mean of $5$ values is $17.$

Here are $4$ of the values:

Find the $5^{th}$ value:

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

The average of $5$ values is $17.$ Multiply these together to find the total of the $5$ numbers.

$\text { Total of } 5 \text { values }=\text { average } \times \text { number of values }=17 \times 5=85$

Find the sum of the known data points.

\text { Total of } 4 \text { values }=4+13+17+21=55

Subtract the sum of the known data points from the first total to find the missing data point.

85-55=30

The $5^{th}$ value is $30.$

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

$\begin{aligned} & \text { average }=\cfrac{\text { sum of values }}{\text { number of values }} \\\\ & 17=\cfrac{4+13+17+21+x}{5} \\\\ & 17=\cfrac{55+x}{5} \\\\ & 85=55+x \\\\ & 30=x \end{aligned}$

Teaching tips for how to find the average in math

Introduce finding the average in math by providing real-life examples that students will naturally use. For example, how students calculate their own grades, the average age of students in their class, or finding the average score on a test.

Simplify the process for students by providing step-by-step instructions, clearly breaking down each step, and allowing students to practice each step before combining them.

Worksheets can be useful for teaching average (mean), but be sure to include a variety of question types – ones with the average missing, ones with a data point missing and a mixture of number types, such as fractions, percentages, decimals and integers.

Easy mistakes to make

Confusing the mean and the median
They are both average values, but do not represent the same average and are found in different ways. The median is the middle number (middle value) when the data set is arranged in ascending (or descending) order. The mean is a number that represents the data points equally shared and is found by adding all the values and then dividing by the number of data points.

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

Thinking the average has to be a number in the data set
While the average can be a number in the data set (group of numbers), often it is not. Adding the data points together and then dividing them by the number of data points allows for the average to be a number outside of the numbers given.

Practice how to find the average in math questions

7

7.2

7.5

6

First calculate the sum of the numbers in the data values.

6+6+7+8+9=36

Then divide the sum by the number of data points.

\begin{aligned} & \text { average }=\cfrac{36}{5} \\\\ & \text { average }=7.2 \end{aligned}

15.9

15.8

15.5

15

First calculate the sum of the values.

12+15+16+19=62

Then divide the sum by the number of data points.

\begin{aligned} & \text { average }=\cfrac{62}{4} \\\\ & \text { average }=15.5 \end{aligned}

16

17

17.5

16.83

\begin{aligned} & \text { average }=\cfrac{\text { sum of values }}{\text { number of values }} \\\\ & \text { average }=\cfrac{22+18+19+16+15+11}{6} \\\\ & \text { average }=\cfrac{101}{6} \\\\ & \text { average }=16.83333 \ldots \\\\ & \text { average }=16.83 \end{aligned}

7, 9, 11, 10, 8

1, 4, 7, 1, 10

5, 6, 9, 2, 5

4, 2, 9, 4, 5

Since each of the given data sets have $5$ data points, the total of the data points will be $9 \times 5$ (the average times the number of data points).

\begin{aligned} & 7+8+9+10+11=45 \\\\ & \text { average }=\cfrac{45}{5} \\\\ & \text { average }=9 \end{aligned}

12

14

16

19

The total of $4$ numbers is:

\text { Total of } 4 \text { values }=\text { average } \times \text { number of values }=13 \times 4=52

The total of $3$ numbers is:

9+10+17=36

The difference between the totals is:

52-36=16

The $4^{th}$ number is $16.$

6

7

12

15

The total of $4$ numbers is:

\text { Total of } 4 \text { values }=\text { mean } \times \text { number of values }=10 \times 6=60

The total of $5$ numbers is:

5+8+11+13+16=53

The difference between the totals is:

60-53=7

The $6^{th}$ number is $7.$

How to find the average in math FAQs

What is the average formula?

The formula for calculating the average, or arithmetic mean, is
$\text { average }=\cfrac{\text { sum of values }}{\text { number of values }}.$

Does the data set of a sample average have to be only positive numbers?

No, the context in which the sample was collected may include negative numbers. For example, temperatures or account balances.

What is a weighted average?

Instead of counting all data points equally, the average of a set is found by counting (or “weighing”) certain data points more than others.

What are other types of mean besides arithmetic?

Geometric mean and harmonic mean are two other types of mean. These are both addressed in upper level mathematics.

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