Median

Order comes first

The median depends on position, so sort the data before looking for the middle. For $9,2,6,4,7$, the ordered list is $2,4,6,7,9$ and the median is $6$.

Unlike the mean, the median does not use the numerical total. It uses rank.

Odd and even counts

With an odd number $n$ of observations, the median is in position $(n+1)/2$. With an even count, average the two central values.

The median need not appear in the data when the count is even.

Interpreting the median

A median of $18$ minutes does not mean every observation is near $18$. It means the ordered data are split around that point. Always pair centre with a spread measure such as range or interquartile range.

The wording “half below and half above” is convenient but can be imprecise when values equal the median. “At or below” and “at or above” handles ties correctly.

Context example

Suppose apartment rents are $1200,1250,1300,1350,6000$. The median is $1300$, while the mean is $2220$. Both calculations are correct, but the median better reflects a typical unit in this small set because one luxury rent pulls the mean.

Common mistakes

Finding the middle before sorting. Position only makes sense in order.

Forgetting to average two central values. Even-sized data sets have two middle positions.

Using frequencies as data values. Frequencies tell how many times values occur.

Claiming the median describes spread. It describes centre; pair it with another statistic.

Quick self-check

• Is the data ordered?
• Is the observation count odd or even?
• Which position or two positions define the middle?
• Would outliers make median more informative than mean?