# Arithmetic mean in statistics

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

Arithmetic mean   =
Sum of all values / Number of values

Symbols used for the mean

It is important to see the difference between the symbols/formula used for a sample data and the symbols/formula used for a population data.

If you compute the mean for a sample data, use the following symbols and formula below:

x for the mean

n for the number of values

Σx for the sum of all values

x   =
Σx / n
If you compute the mean for a population data, use the following symbols and formula below:

µ for the mean

N for the number of values

Σx for the sum of all values

µ   =
Σx / N

## Examples showing how to find the arithmetic mean in statistics

The yearly incomes (in thousands of dollars) of all 10 employees of a small company are

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

Find the mean.

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

µ   =
Σx / N

µ   =
36 + 45 + 500 + 30 + 40 + 50 + 45 + 40 + 48 + 55 / 10

µ   =
889 / 10

µ = 88.9

The mean for the population is 88.9 thousand

Notice that 500 is high compared to the other values. What will happen if we remove it and compute the mean again?

µ   =
36 + 45 + 30 + 40 + 50 + 45 + 40 + 48 + 55 / 9

µ   =
389 / 9

µ = 43.22

The mean for the population is 43.22 thousand

We see that when 500 is included in the data set, the mean is twice as big.

If this company is trying to hire people and says in an advertisement that the average income is 88.3 thousand, it may give the wrong information to those who are looking for a job.

People looking for a job may wrongly assume that they will earn 88.3 thousand dollars. However, it is quite possible that it is the owner of the company who is making 500 thousand.

500 is an outlier because it changes the mean drastically. Values  that increase or decrease the mean drastically are called outliers. Always be cautious when interpreting the mean. Because it can be influenced by outliers, it is not always a good measure of central tendency

You are free to collect a sample from the set to find the mean.

x   =
Σx / n

x   =
30 + 30 + 40 / 3
= 33.33

x   =
50 + 500 + 55 / 3
= 201.66

See the striking difference! This too illustrate the need to be careful too when choosing the sample.

Furthermore, the value of the sample mean can change while the value of the population is always the same.

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