Statistics Foundations

Mean, Median, and Mode

Three ways to describe the center of a data set.

Foundational20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Three Different 'Averages' — and Why They Can Disagree

A small company's employees earn these yearly salaries, in thousands: 45, 48, 50, 52, 200 — that last number belonging to the company's founder. Before reading on, think about which single number would honestly describe a 'typical' salary at this company, and whether the ordinary average is really the best choice here.

Definition — Mean, Median, and Mode

The mean is the sum of all values divided by how many there are. The median is the middle value once the data is sorted. The mode is the value that occurs most often.

The mean of the salary data is (45+48+50+52+200) ÷ 5 = 79 — a number no actual employee besides the founder comes anywhere close to earning. The single extreme value pulled the mean far higher than what's typical. The median, though, is unaffected by exactly how extreme that outlier is — it only cares about the value's position once sorted, not its size.

Worked Example — Comparing Mean and Median with an Outlier

Find the median of the salary data: 45, 48, 50, 52, 200 (already sorted). The middle value is 50. Compare this to the mean, 79 — the median, 50, is a much more honest description of a 'typical' salary here, since it isn't dragged upward by the one extreme value.

Worked Example — Finding the Mode of a Data Set

A shoe store's sales for the day, by size: 8, 9, 9, 10, 9, 11, 8. Which size sold most often? Size 9 appears three times, more than any other size, so the mode is 9.

Tip

When a data set contains an extreme outlier, the median usually gives a more honest sense of 'typical' than the mean — checking both, and comparing how far apart they land, quickly reveals whether an outlier is distorting the mean.

Common Mistakes

Finding the median from unsorted data, picking the value that happens to sit in the middle position of the original, unsorted list.
Sort every value from least to greatest first — only then does the middle position actually represent the median.
Assuming the mean is always the best measure of 'typical,' even when a data set contains an extreme outlier.
Check whether a data set has values far away from the rest — an extreme outlier pulls the mean toward it much more than it affects the median.

Key Takeaways

The mean, median, and mode each describe a data set's center differently, and can genuinely disagree with each other.
The median resists the pull of extreme outliers far better than the mean does.
The mode identifies the single most frequently occurring value, and is most useful for data that isn't purely numerical.

Summary

Choosing between mean, median, and mode means understanding what each one is actually sensitive to. The next lesson shifts from describing data that's already happened to predicting the likelihood of events that haven't happened yet.

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