Mean (average)

What the symbols mean

$\overline{x}$

the mean of the data ($\mu$ is used for a whole population)

$x_i$

the data values $x_1,x_2,\dots ,x_n$

$n$

the number of values

$\sum$

"add up": the sum of all the values

When to use it

Use the mean for a typical value when the data are roughly symmetric: test scores, heights, daily temperatures. If a few values are extreme (incomes, house prices), the median is often a better summary, because one very large value pulls the mean a long way. The calculator shows both. Type the data as numbers separated by commas or spaces.

Why it works

The mean is the balance point of the data: the distances above it exactly cancel the distances below it, so $\sum (x_i-\overline{x})=0$. It is also the single number that makes the sum of squared distances as small as possible, which is why the standard deviation is measured from it.

Common mistakes

• Dividing by the wrong count, for example forgetting a zero score, which still counts as a value.
• Averaging averages: the mean of two class means is not the overall mean unless the classes are the same size.
• Using the mean for skewed data, where it can be far from any typical value.