The **arithmetic mean** is the most commonly used measure of central tendency in statistics. You can only find the arithmetic mean for quantitative data. To find it, use the following formula:

Arithmetic mean =

Sum of all values

Number of values

**Symbols for the mean value**

It is important to recognize the difference between the symbols / formulas used for a **Sample data** and the symbols / formulas used for a **Population data**.

**sample**Data, use the following symbols and formulas below:

x for the mean

n for the number of values

Σx for the sum of all values

**population**Data, use the following symbols and formulas below:

µ for the mean

N for the number of values

Σx for the sum of all values

## Examples showing how the arithmetic mean is found in statistics

The annual income (in thousands of dollars) of **all** 10 are employees of a small business

36, 45, 500, 30, 40, 50, 45, 40, 48 and 55.

Find the mean.

Since the data is given for all 10 employees, the mean we are looking for is the population mean.

µ =

36 + 45 + 500 + 30 + 40 + 50 + 45 + 40 + 48 + 55

10

µ = 88.9

The mean value for the population is 88.9 thousand

Note that 500 is high compared to the other values. What if we remove it and recalculate the mean?

µ =

36 + 45 + 30 + 40 + 50 + 45 + 40 + 48 + 55

9

µ = 43.22

The mean value for the population is 43.22 thousand

We see that the mean is twice as large if there are 500 in the data set.

If this company is trying to hire people and an ad says the average income is 88.3 thousand, it may be giving the wrong information to those looking for a job.

People looking for a job may mistakenly assume that they will make 88.3 thousand dollars. However, it is entirely possible that the owner of the company will make $ 500,000.

500 is a **Runaway** because it changes the mean drastically. Values that dramatically increase or decrease the mean are known as outliers. Always be careful when interpreting the mean. Since outliers can affect it, it’s not always a good measure of the central bias

You are free to take a sample from the set to determine the mean value.

30 + 30 + 40

3

= 33.33

50 + 500 + 55

3

= 201.66

See the noticeable difference! This also shows that caution should also be exercised when selecting the sample.

In addition, the sample mean value can change while the population value is always the same.