Mean, Median, Mode & Range

Four ways to describe a list of numbers:

• Mean: add them all, divide by how many. The everyday "average."
• Median: sort them and take the middle one. With an even count, average the middle two.
• Mode: the value that appears most often. There can be none, or several.
• Range: largest minus smallest. A measure of spread, not centre.

Sorting first is mandatory for the median, and it is the step most often skipped.

The mean gets dragged around by extreme values. One billionaire in a room of ten people makes the mean income meaningless while the median stays honest.

The mean x̄ = (Σxᵢ)/n is the unique value at which deviations balance, since Σ(xᵢ − x̄) = 0. That balancing property is exactly why it responds to every value, including outliers, and why it is the right centre for further statistical work.

The median is the 50th percentile and is resistant: changing an extreme value arbitrarily does not move it. Its breakdown point is 50%, against 0% for the mean. This is why income and house-price statistics are reported as medians while means are used for symmetric measurements.

The relationship between them diagnoses shape. Mean above median suggests a right skew, mean below suggests left skew, and rough equality suggests symmetry.

Range uses only two data points and is therefore a fragile measure of spread; interquartile range (Q3 − Q1) and standard deviation are the resistant and the standard alternatives respectively.