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Multiplication and division Decimals Rounding numbersHere you will learn about the mean, including what the mean is and how to find the mean.

Students will first learn about the mean in math as part of statistics and probability in 6 th grade.

The **mean in math**, specifically the arithmetic mean, is a type of average calculated by finding the total of the values and dividing the total by the number of values.

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}

For example,

Calculate the mean of 3, \, 8, \, 10, \, 11 and 13.

\text {Mean }=\cfrac{\text { total }}{\text { number of values }}=\cfrac{3+8+10+11+13}{5}=\cfrac{45}{5}=9

9 is the mean of the data set.

This value, also known as the population mean, is a measure of central tendency. It summarizes a data set (population) with a single point. 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 example,

It is also important to consider that the number of observations (number of data points) changes how much each data point affects the mean.

For example,

Data set A\text{: } 2, 3, 10 | Data set B\text{: } 2, 2, 3, 3, 10, 10 |

Mean: 15 \div 3 = 5 | Mean: 30 \div 6 = 5 |

Now add the data point 10 to each data set and recalculate the mean.

Data set A\text{: } 2, 3, 10, 10 | Data set B\text{: } 2, 2, 3, 3, 10, 10, 10 |

Mean: 25 \div 4 = 6.25 | Mean: 40 \div 7 = 5.7 |

Notice that the mean in data set A grew by 1.25, while the mean of data set B grew by 0.7.

Since there are less data points in A , adding (or taking away) a data point impacts the mean more than in a data set with more points, like data set B.

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 calculate the mean in math:

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

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEAssess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREECalculate the mean value of this list of numbers:

2 \quad 7 \quad 9 \quad 10 \quad 12

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

2+7+9+10+12=40

2**Divide the sum by the number of data points.**

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

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{40}{5}=8

3**Write down the answer.**

The mean is 8.

Calculate the mean value of this set of numbers to the nearest tenth.

13 \quad 16 \quad 17 \quad 17 \quad 18 \quad 20

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

13+16+17+17+18+20=101

**Divide the sum by the number of data points.**

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

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{101}{6}=16.8333…

**Write down the answer.**

16.8333… to the nearest tenth is 16.8.

16.8 is the mean.

Calculate the mean value of this set of data to the nearest hundredth.

11 \quad 13 \quad 14 \quad 15 \quad 19 \quad 20 \quad 22

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

11+13+14+15+19+20+22=114

**Divide the sum by the number of data points.**

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

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{114}{7}=16.2857…

**Write down the answer.**

16.2857… rounded to the nearest hundredth is 16.29.

16.29 is the mean.

Calculate the mean value of this list of numbers.

101 \quad 102 \quad 105 \quad 106 \quad 108

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

101+102+105+106+108=522

**Divide the sum by the number of data points.**

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

\text{Mean}=\cfrac{\text{total}}{\text{number of values}}=\cfrac{522}{5}=104.4

**Write down the answer.**

104.4 is the mean.

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

**Use the mean 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 mean of 4 values is 10.

Here are 3 of the values:

6 \quad 9 \quad 12

Find the 4^{th} value.

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

The mean of 4 values is 10. Multiply these together to find the total of the 4 numbers.

\text{Total of 4 values}=\text{mean} \times \text{number of values}=10\times 4=40

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

\text{Total of 3 values}=6+9+12=27

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

40-27=13

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.

\begin{aligned} \text{Mean} &= \cfrac{\text{total}}{\text{number of values}}\\\\ 10&=\cfrac{6+9+12+x}{4}\\\\ 10 &= \cfrac{27+x}{4}\\\\ 40&=27+x\\\\ 13&=x \end{aligned}

The mean of 5 values is 14.

Here are 4 of the values:

5 \quad 11 \quad 13 \quad 19

Find the 5^{th} value:

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

The mean of 5 values is 14. Multiply these together to find the total of the 5 numbers.

\text{Total of 5 values}=\text{mean} \times \text{number of values}=14\times 5=70

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

\text{Total of 4 values}=5+11+13+19=48

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

70-48=22

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.

\begin{aligned}
\text{Mean} &= \cfrac{\text{total}}{\text{number of values}}\\\\
14&=\cfrac{5+11+13+19+x}{5}\\\\
14 &= \cfrac{48+x}{5}\\\\
70&=48+x\\\\
22&=x\\\\
\end{aligned}

- Introduce mean with countable objects (like counters or connecting cubes). This will allow students to physically add all the data points together and then share them equally – matching the procedure for the mean. Doing a hands-on activity like this helps students make sense of what the mean represents and understand why the procedure to find mean works.

- Worksheets can be useful for teaching mean, but be sure to include a variety of question types – ones with mean missing, ones with a data point missing and a mixture of number types, such as fractions, 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 mean 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.

**Thinking the mean has to be a number in the data set**

While the mean can be a number in the data set, often it is not. Adding the data points together and then dividing them by the number of data points allows for the mean to be a number outside of what is in the data set.

- Averages and range
- Median
- Mean median mode
- Mode in math
- Range in math

1) Find the arithmetic mean of this set of values:

5 \quad 7 \quad 8 \quad 8 \quad 9

7

7.4

7.5

8

First calculate the sum of the numbers in the given set.

5+7+8+8+9=37

Then divide the sum by the number of data points.

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{5+7+8+8+9}{5}\\\\ \text{Mean}&= \cfrac{37}{5}\\\\ \text{Mean}&=7.4\\\\ \end{aligned}

2) Find the mean of this list of values:

3 \quad 5 \quad 6 \quad 9

5.9

5.8

5.75

5.7

First calculate the sum of the values.

3+5+6+9=23

Then divide the sum by the number of data points.

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{3+5+6+9}{4}\\\\ \text{Mean}&= \cfrac{23}{4}\\\\ \text{Mean}&=5.75\\\\ \end{aligned}

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

7 \quad 8 \quad 9 \quad 10 \quad 10 \quad 11

9.16

9.5

10

9.17

\begin{aligned} \text{Mean}&= \cfrac{\text{total}}{\text{number of values}}\\\\ \text{Mean}&=\cfrac{7+8+9+10+10+11}{6}\\\\ \text{Mean}&= \cfrac{55}{6}\\\\ \text{Mean}&=9.16666…\\\\ \text{Mean}&=9.17\\\\ \end{aligned}

4) Which data set has a mean of 6?

1, \, 15, \, 2, \, 1, \, 17, \, 0

1, \, 4, \, 7, \, 1, \, 10, \, 15

5, \, 6, \, 9, \, 6, \, 2, \, 5

4, \, 8, \, 2, \, 9, \, 4, \, 5

Since each of the given data sets have 6 data points, the total of the data points will be 6 \times 6 (the mean times the number of data points).

\begin{aligned} & 1+15+2+1+17+0=36 \\\\ & \text { Mean }=\cfrac{36}{6} \\\\ & \text { Mean }=6 \end{aligned}

5) The mean of 4 numbers is 9.

Here are 3 of the numbers:

6 \quad 8 \quad 15

What is the 4^{th} number?

5

6

7

8

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+15=29

The difference between the totals is:

36-29=7

The 4^{th} number is 7.

6) The mean of 6 numbers is 12.

Here are 5 of the numbers:

7 \quad 9 \quad 11 \quad 13 \quad 18

What is the 6^{th} number?

12

14

13

15

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

The 6^{th} number is 14.

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 mean of a set is found by counting (or “weighing”) certain data points more than others.

A way to quantify the amount of variation around the mean within a data set or population.

In a normal distribution, all the measures of center are the same and are exactly at the center value. Thinking about the data in percentages, 68\% of the points are within one standard deviation of the mean, median and mode.

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

- Frequency table
- Cumulative frequency
- Types of data
- Representing data
- Units of measurement

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