Statistics

Centre, spread, and shape.

Descriptive statistics answer three questions about a data set: where is its centre, how spread out is it, and what shape does it have. Nearly every summary figure is a different answer to one of those three.

Centre

The mean is the arithmetic average and uses every value, which makes it sensitive to outliers. The median is the middle value in rank order and ignores how extreme the extremes are. The mode is the most frequent value and is the only one that works for categories. When a distribution is skewed, the median describes a typical case better than the mean.

Spread

The range is the largest value minus the smallest and is decided entirely by two observations. The interquartile range covers the middle half and resists outliers. The standard deviation reports typical distance from the mean in the original units.

Shape

A symmetric distribution has mean and median close together. In a right-skewed distribution a long tail of high values pulls the mean above the median; income data behaves this way. Reporting only a mean for skewed data reliably misleads.