What Is Arithmetic Mean?

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To find the mean or average of a list of numbers, you add all the numbers together and divide by the total number of observations.

Rule

Mean

You can find the mean of a list of number by adding all the numbers you have together, and then divide that sum by the number of observations you have.

Mean = \frac{o_{1} + o_{2} + \dots + o_{n}}{number of observations}

where $o_{1}$ is observation number $1$ , $o_{2}$ is observation number $2$ , etc.

A more theoretical approach to the mean gives us this formula:

Formula

Mean: Non-Grouped Data

A mean is the sum of the $n$ observational values in a set of data points divided by the number $N$ of values in the set of data points. The mean is written symbolically as $\overset{}{x}$ (pronounced “x bar”).

\overset{}{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{N}

Example 1

You asked eight students in your class how many minutes they spent doing homework, and got these answers:

10 17 6 20 5 10 5 19

Find the mean.

First you add all the numbers together:

\begin{array}{l} 10 + 17 + 6 + 20 \\ + & 5 + 10 + 5 + 19 = 92 \end{array}

10 + 17 + 6 + 20 + 5 + 10 + 5 + 19 = 92

You must divide this sum by the number of values you have in the list:

92 \div 8 = 11.5

The mean is $11.5$ minutes.

Think About This

What does it mean that the mean is $11.5$ ? What is the range of variation? Where in the observations is the mean?

That is, if the time everyone spent on homework was distributed equally, everyone would spend $11.5$ minutes on it. The range is $20 - 5 = 15$ minutes. The distance from the mean to the maximum is $20 - 11.5 = 8.5$ . The distance from the mean to the minimum is $11.5 - 5 = 6.5$ . From this, you see that the mean is in the lower part of the data set, since the mean is closer to the minimum value.

Think About This

What do you think happens to the mean when the data set has extreme values?

When the data set has extreme values, this value will pull the mean value sharply in the direction of those values. This can give a skewed picture of the information you have collected. In such cases, it may be wise to use one of the other measures of central tendency together with the mean—or alone—to show a more accurate picture of the reality.

Example 2

Find the mean age of the members of the Upper West Side Gymnastics Group. The group is made up of 15 gymnasts with the following ages: 4, 5, 6, 3, 6, 12, 12, 14, 15, 13, 12, 12, 13, 14, 15.

Insert the numbers into the formula and solve

\begin{array}{l} \overset{}{x} & = \frac{1}{15} \cdot (4 + 5 + 6 + 3 + 6 \\ + 12 + 12 + 14 + 15 + 13 \\ + 12 + 12 + 13 + 14 + 15) \\ = \frac{156}{15} \\ = 10.4 years \end{array}

\begin{array}{l} \overset{}{x} & = \frac{4 + 5 + 6 + 3 + 6 + 12 + 12 + 14 + 15 + 13 + 12 + 12 + 13 + 14 + 15}{15} \\ = \frac{156}{15} \\ = 10.4 years \end{array}

The mean age of the group is

10.4

years.

Example 3

You have a list of the heights of the 14 players on the handball team measured in meters.

Height $1.75$ $1.80$ $1.82$ $1.79$
$1.75$ $1.89$ $1.92$ $1.82$ $1.83$
$1.82$ $1.90$ $1.89$ $1.79$ $1.85$

Find the mean height of the team.

\begin{array}{l} \overset{}{x} & = \frac{1}{14} \cdot (1.75 + 1.80 + 1.82 + 1.79 + 1.75 \\ + 1.89 + 1.92 + 1.82 + 1.83 + 1.82 \\ + 1.90 + 1.89 + 1.79 + 1.85) \\ = \frac{25.62}{14} \\ = 1.83 \end{array}

The mean height of the team is

1.83

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How to Find Mean of a Grouped Frequency Table

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Height	$1.75$	$1.80$	$1.82$	$1.79$

$1.75$	$1.89$	$1.92$	$1.82$	$1.83$

$1.82$	$1.90$	$1.89$	$1.79$	$1.85$