Median — Individual, Discrete and Continuous Series

The median is the value that divides an array of data into two exactly equal halves — half the observations lie below it and half lie above it, once the data is arranged in ascending (or descending) order of magnitude. Unlike the mean, the median does not depend on the actual magnitude of every value, only on relative position — which is exactly why an extreme value barely moves it (Question 14's MCQ compares this directly against the mean).

Individual series. Arrange the $n$ observations in ascending order.

• If $n$ is odd, the median is the value at position $\dfrac{n+1}{2}$.
• If $n$ is even, the median is the arithmetic mean of the values at positions $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$.

Discrete series. Arrange the values in ascending order and build the cumulative frequency (cf) column. Locate $\dfrac{N}{2}$ (where $N = \Sigma f$) in the cf column: the first value of $x$ whose cumulative frequency is equal to or just greater than $\dfrac{N}{2}$ is the median.

Continuous series. First identify the median class — the class whose cumulative frequency is equal to or just exceeds $\dfrac{N}{2}$ — then interpolate within it:

where $L$ is the lower limit of the median class, $cf$ is the cumulative frequency of the class immediately before the median class, $f$ is the frequency of the median class itself, and $h$ is the class width. …