Mean (Average)

The arithmetic mean of $x_1, x_2, \ldots, x_n$ is

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.$

It is the value that minimises the sum of squared deviations $\sum (x_i - c)^2$ — that is why squared loss is everywhere in statistics and machine learning: minimising squared loss means estimating means.

The mean is sensitive to outliers: a single extreme value can pull the mean far from where most data lies. When data is skewed (income, response time, file size) the median is often a better summary. Other averages — geometric, harmonic, weighted — apply in specific contexts (compounded growth, parallel resistors, weighted polls).