Mean (average)

The mean is a value often used in statistics, worked out as the arithmetic mean of a set of values.

What is the mean

By the mean we usually understand the ordinary arithmetic mean of the values we are looking at. As an example, take this table of monthly salaries in pounds:

These are the salaries of 11 imaginary employees of our imaginary company, Matweb Ltd. What would the mean salary in this company be? We take all the amounts and add them up:

$$\begin{array}{l}\mathrm{1,000}+\mathrm{1,500}+\mathrm{1,900}+\mathrm{2,100}+\mathrm{1,800}+\\ \mathrm{2,500}+\mathrm{1,500}+\mathrm{4,500}+\mathrm{2,900}+\mathrm{8,500}+\mathrm{1,500}=\mathrm{29,700}\end{array}$$

In total, £29,700 a month is paid out in salaries. We get the mean salary by dividing this amount by the number of employees, that is, eleven: 29,700 / 11 = 2,700. The mean salary in our company is £2,700. Not bad, is it?

So if we have a data set of n values, which we call x₁, x₂, …, x_n, we get the mean by adding up all the x_i and dividing the result by n. The mean is usually written $\bar{x}$. The formula:

$$\bar{x}=\frac{x_1+x_2+\dots +x_n}{n}$$

Or the same formula using sigma notation:

$$\bar{x}=\frac{1}{n}\cdot \sum _{i=1}^n{x}_i$$

How to work out the mean in Excel

In Excel, the mean is worked out with the AVERAGE function. As its argument, enter the range of cells you want to find the mean of.

Problems with the mean

Some problems you may run into with the mean:

• The mean is neither the most common nor the most likely value. It is simply the mean. If we worked out the average of school grades, we might end up with 6.4, which is not even a valid grade you could get at school. But it is the average grade. In the same way, the mean salary in our company is £2,700, yet nobody actually earns that amount.
• The mean can give a misleading picture of the data if the data contain some large extreme values. A mean salary of £2,700 sounds quite nice, but most employees come nowhere near it. Such a high mean is mainly due to a single employee, Edward, who earns £8,500 and pulls the mean up a lot. If we sacked Edward, the mean salary would be only £2,120, which at first glance looks like a less misleading figure.