Mean, median and mode: choosing a summary statistic that does not mislead

What it is

The NIST/SEMATECH handbook defines three measures of location: the mean is the sum of the data points divided by their number; the median is the value of the point which has half the data smaller than it and half larger; the mode is the value that occurs most often, is not necessarily unique, and is typically used in a qualitative fashion. It also lists alternatives to the mean and median that were developed for non-normal data: the mid-mean (mean of the values between the 25th and 75th percentiles), the trimmed mean and the winsorized mean. The Python statistics documentation states that the mean is strongly affected by outliers and is not necessarily a typical example of the data points, that the median is a robust measure of central location and is less affected by outliers, and that the harmonic mean is often appropriate when averaging ratios or rates such as speeds.

Why it matters

Each summary answers a different question. "What will a thousand such items cost in total?" needs the mean, because mean times count equals the sum. "What does a typical item look like?" wants the median whenever the distribution is skewed, which durations, sizes and costs nearly always are. Reporting the mean of a skewed distribution as "typical" overstates the typical case; reporting the median as a planning figure understates the total.

How to apply

• Look at the shape before summarising: a histogram or the sorted values. Symmetric with light tails: mean and median agree, either works. Skewed or heavy-tailed: report the median and a high percentile, and the mean only when totals matter.
• For ratios and rates (requests per second, kilometres per hour), compute the ratio of the sums, or use the harmonic mean when every ratio deserves equal weight; the arithmetic mean of ratios ignores the denominators.
• For multiplicative quantities (growth factors, speed-ups), use the geometric mean.
• Never average medians or percentiles across groups or time windows; recompute from the pooled raw data or from histograms.
• Print the count next to every summary; a median of three values is barely a summary.
• Use the mode for categorical data (the most common error class), not for continuous measurements, where it depends on the binning.

Pitfalls

A mean that moves while the median stays still is a tail change, not a change in the typical case, and the reverse. A trimmed mean hides the tail the reader may care about; say what was trimmed. Two groups with equal means can have entirely different shapes. A summary without a spread and a count is half a number.