Premathematically, the mean (German Mittel or Mittelwert) of a set of numerical values is a kind of average value. For instance, the average age of a group of persons may be calculated as the mean of the ages of the individuals. The mean is the most basic of the statistical parameters. Given that each of the values is conventionally designated by x, the mean of the set is represented by x̄.¹

Given a set of n real numbers, then there are a variety of ways of calculating their average value. The median and the mode will be left out of consideration. There remain three established ways of defining the mean: the arithmetic mean, the geometric mean and the harmonic mean.

• The arithmetic mean is the default that is applicable in all cases where neither of the other two means is required.
• The geometric mean is needed if the data are skewed, which is typically the case if the values are rates of growth.
• The harmonic mean is needed if the values add up with different weights to a whole, as is the case, e.g., if a distance is divided into a series of chunks each of which is covered with some different speed, and the average speed over the distance is to be calculated.

Here it will suffice to consider the arithmetic mean.

The arithmetic mean x̄a of a set of values { x1 ... xn } is the sum (∑) of these values divided by n. In a formula:

This may be illustrated by a sample that comprises the values { 2.0, 4.0, 4.0, 5.0 }; thus, n is 4. For this example set, the arithmetic mean is:
(2.0 + 4.0 + 4.0 + 5.0) / 4 = 15.0 / 4 = 3.75.

For many elementary statistical purposes, it may seem sufficient to indicate the mean of a set of numerical values. However, in order to be meaningful, a mean must always be accompanied by the standard deviation of the distribution.

¹ Alternatively, the symbol μ is used; but this should probably be reserved for the estimated mean of the population.