Measures of Center (Mean, Median, Mode)

The four measures

Mean: add every value, then divide by how many values there are. For $4, 9, 5, 6$ the mean is $\dfrac{4 + 9 + 5 + 6}{4} = \dfrac{24}{4} = 6$. Forgetting the divide step — reporting the total instead of the mean — is the classic mistake.

Median: put the values in order first, then take the middle one. Ordering is not optional; the median of an unordered list means nothing.

Mode: the most frequent value. A data set can have one mode, more than one mode, or none at all if every value appears the same number of times. It is the only measure of center that requires no arithmetic — just counting.

Range: greatest value minus least value. It says nothing about the center; it measures how far the data stretches.

The median with an even count

With an odd number of values, one value sits exactly in the middle. Five values ordered? The third one is the median — two on each side.

With an even number of values, no single value is in the middle, so the median is the mean of the two middle values. For the ordered set $6, 8, 10, 14, 18, 22$, the two middle values are $10$ and $14$, so the median is $\dfrac{10 + 14}{2} = 12$. Notice the median does not have to be a value in the data set.

Which measure should you use?

The mean uses every value, which makes it the most informative — and the most fragile. One extreme value (an outlier) drags the mean toward it. In the set $2, 3, 3, 4, 88$, the mean is $20$, but the median is $3$, which describes the typical value far better.

Use the median when the data has outliers or is skewed. Use the mode when the data is categories or when you want the most common result. When a test question says average, it almost always means the mean.

Worked examples

Try one yourself

Common questions