Median

The median is the middle value of an ordered data set, with half the observations at or below it and half at or above it.

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.

Why the median resists outliers

Changing the largest value from $20$ to $2000$ leaves the previous list’s median at $9.5$. The order of the middle positions does not change. The mean, however, increases dramatically.

This resistance makes median useful for skewed data such as income, home price, or waiting time.

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.

Frequency tables

Use cumulative frequency to locate the middle position without listing every observation. If $31$ responses are recorded, the median is the $16$th ordered response. Add frequencies until the cumulative count reaches that position.

For grouped continuous data, the exact median may not be recoverable; a median class or interpolated estimate may be reported.

Median and quartiles

The median is the second quartile, $Q_2$, and the $50$th percentile. Quartiles extend the same ordered-position idea: $Q_1$ describes the lower half and $Q_3$ the upper half. Conventions for including the overall median when splitting an odd-sized set can differ, so state the course convention.

Choosing mean or median

Use the median when a distribution is strongly skewed, contains important outliers, or uses ordinal ranks. Use the mean when all values should influence the balance point and the distribution is reasonably symmetric. Reporting both can reveal skew: a mean far above the median often signals a long right tail.

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?

Related topics

• Mean
• Mode
• Range
• Box Plots