Three ways to say 'typical'
The mean adds everything and divides by the count: every value votes with its size. The median is the middle value once sorted: every value votes with its position only. The mode is the most frequent value. They are three different definitions of centre, and on real data they disagree informatively: the pattern of disagreement diagnoses the distribution's shape.
A batsman's nine innings: 45, 12, 78, 33, 45, 0, 56, 45, 21. Mean 37.2, median 45, mode 45. The mean sits below both because the 0 and 12 drag it down while the 78 cannot fully compensate. Median and mode tell the everyday story (a typical innings is in the forties); the mean quietly includes the failures at full weight. Neither is lying; they answer different questions.
Choosing by data shape
Symmetric data: all three huddle together; use the mean, it uses every value's information and pairs naturally with the standard deviation. Skewed data or outliers: the median leads, with quartiles as its spread; the mean still gets reported when totals matter (mean × n is the total, which budgets and production plans need). Categorical data: only the mode exists (the most common category); averaging category codes is arithmetic without meaning.
Reporting only the mean for income-like data. Eight salaries (in thousands) 32, 35, 38, 40, 42, 45, 48, 240: mean 65, median 41. "Average pay is 65k" describes a workplace where seven of eight people earn under 48k. The median is the honest headline; the mean belongs in the payroll budget, where the total is what counts.
The mode's special cases
Data can have no mode (all values unique) or several (bimodal: two humps, often two mixed groups). A bimodal report should trigger investigation, not a coin flip between modes: find the grouping variable. Continuous measurements rarely repeat exactly, so modes belong mostly to counts and categories.
Frequently asked questions
Which measure of central tendency is best?
The one matched to the data: mean for symmetric measurements (and whenever totals matter), median for skewed data and outliers, mode for categories and frequency questions. Reporting mean and median together covers most situations honestly.
Why does the mean use every value but the median does not?
By construction. The mean is a total shared equally, so each value pulls it in proportion to its size. The median depends only on order: the extreme values know their rank and nothing else, which is exactly its protection against outliers.
What does it mean when mean, median and mode are equal?
The distribution is symmetric and single-peaked around that value (the normal distribution is the famous case). It is a useful quick diagnostic: equality of the three is a signature of symmetry.