Arithmetic mean

The mean is the sum of the values divided by their number — the balance point of the data, sensitive to every value including outliers.

\xbar = \frac{1}{n}\sum_{i=1}^{n} x_i \qquad \mu = \frac{1}{N}\sum_{i=1}^{N} x_i

Sample mean and population mean.

$\xbar$: Sample mean.
$\mu$: Population mean.
$x_i$: The $i$ -th value.
$n,\ N$: Sample size and population size.

When to use

Quantitative (interval or ratio) data, especially roughly symmetric distributions.
When totals matter: mean × n recovers the total (total sales, total minutes).
Not for nominal data, and use with care for ordinal codes (e.g. Likert 1–5).

How to interpret

The mean is the value every observation would have if the total were shared equally. Deviations from it sum to zero. In a right-skewed distribution the mean is pulled above the median; it is then not "the typical value".

Common mistakes

Reporting only the mean for skewed data (incomes, waiting times, claim sizes) — add the median.
Averaging percentages or rates from groups of different sizes with equal weights (use a weighted mean).
Averaging growth rates arithmetically (use the geometric mean).

Example

Values 4, 7, 7, 9, 13: $\xbar = 40/5 = 8$ , median 7. Replace 13 by 63: the mean jumps to 18 while the median stays 7 — one extreme value moved the mean by 10.