The same data can look typical in very different ways depending on which center you choose.
Suppose you need to summarize the salaries of nine employees with one number. If eight salaries are similar but one is much higher, the mean salary may be far above what most employees actually earn.
A measure of center compresses the data into one value, but the story changes with the measure you choose. The goal is not to use the mean automatically, but to choose a center that fits the measurement scale and the shape of the distribution.
Key question
Which measure—the mean, median, or mode—best represents a typical value in these data?
How do the three measures differ?
| Measure | Calculation or definition | Strength |
|---|---|---|
| Mean | add all values and divide by the count | uses the magnitude of every value |
| Median | middle value after sorting | relatively resistant to extremes |
| Mode | most frequent value | also works for categorical data |
Arithmetic mean
x̄ = (x₁ + x₂ + ··· + xₙ) / n
For example, the mean of 60, 70, and 80 is (60+70+80)/3=70. You can think of the mean as redistributing the total equally across all observations.
Why does the mean move with extremes?
For 60, 70, and 80, both the mean and median are 70. If the last value becomes 800, the mean rises to 310 while the median remains 70. The mean responds to magnitude; the median depends on ordered position.
Income data
For income, home prices, or waiting times with a long right tail, the median may better describe a typical person or case.
The mean is not inherently bad
If an extreme value is valid and substantively important, allowing it to influence the mean may be exactly what you want.
When are the median and mode useful?
To find the median, sort the values and take the middle one. With an even number of observations, use the average of the two middle values. The mode is the most frequent value, and a data set may have no mode or more than one.
| Data type | Available measures of center |
|---|---|
| Blood type or preferred brand | Mode |
| Satisfaction rating from 1 to 5 | Median and mode |
| Height, score, or sales | Mean and median; mode if values repeat |
You do not have to choose only one
For skewed data or data with extremes, reporting both the mean and median can reveal the center and the skew at the same time.
Which measure should you choose?
| Situation | First measure to inspect | Reason |
|---|---|---|
| Roughly symmetric continuous data | Mean | uses the information in every value |
| Skewed data with extreme values | Median | less pulled by a few large values |
| Most common category matters | Mode | identifies the most frequent category |
| Comparing distribution shape | Mean and median together | their difference can signal skew |
Common misinterpretation
A mean of 70 does not imply that most observations are clustered near 70. A measure of center should be read together with spread and distribution shape.
Move an extreme value
First keep the last observation close to the others and compare all three measures. Then increase it sharply and see which moves more—the mean or the median.
Mean
5.44
Median
5.00
Mode
5.00
As this value moves away from the rest, the mean changes the most.
What to notice
As the extreme value grows, the mean moves quickly. The median barely changes unless the ordering changes, and the mode changes only when the repeated value changes.