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Addition and subtraction Multiplication and division Decimals Rounding numbersHere 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.

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.

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}=1212 is the average of the data set.

Use this quiz to check your 6th grade students’ understanding of averages and range. 10+ questions with answers covering a range of 6th grade averages and range topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your 6th grade students’ understanding of averages and range. 10+ questions with answers covering a range of 6th grade averages and range topics to identify areas of strength and support!

DOWNLOAD FREEThere 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.

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.**

Calculate the average of this list of numbers:

**Find the sum of the data points.**

2**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 the values }}{\text { number of values }}=\cfrac{50}{5}=103**Write down the answer.**

The average is 10.

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.

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.

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.

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.**

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}

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}

- 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.

**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.

1. Find the average of this set of values:

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}

2. Find the average of this set of values:

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}

3. Find the average of this list of values. Round your answer to the nearest hundredth.

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}

4. Which data set has an average of 9?

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}

5. The average of 4 numbers is 13.

Here are 3 of the numbers

What is the 4^{th} number?

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. The mean of 6 numbers is 10.

Here are 5 of the numbers

What is the 6^{th} number?

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.

The formula for calculating the average, or arithmetic mean, is

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

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

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

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

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