Median — Computation, Merits and Limitations

Median is the value of the middle-most observation when the data is arranged in ascending (or descending) order — it divides the distribution into two exactly equal halves, half the observations lying below it and half above it.

1. Individual series: first arrange the data in ascending order, then:

• If $n$ is odd, Median = value of the $\left(\dfrac{n+1}{2}\right)^{th}$ observation.
• If $n$ is even, Median = average of the $\left(\dfrac{n}{2}\right)^{th}$ and $\left(\dfrac{n}{2}+1\right)^{th}$ observations.

2. Discrete series: construct a cumulative frequency (c.f.) column, locate $\dfrac{N}{2}$, and find the FIRST cumulative frequency that is greater than or equal to $\dfrac{N}{2}$ — the $x$ value against that cumulative frequency is the median. (If $N$ is even and $\dfrac{N}{2}$ itself exactly equals a cumulative frequency, the median is the average of that value and the very next value — the same even/odd logic as individual series, applied through the c.f. column.)

3. Continuous series: locate the median class — the class in which the cumulative frequency first reaches or exceeds $\dfrac{N}{2}$ — then apply:

where $L$ = lower limit of the median class, $cf$ = cumulative frequency of the class PRECEDING the median class, $f$ = frequency of the median class, and $h$ = class width.

Merits of median:

• Not affected by extreme values (outliers) — an ideal measure when a distribution has a few unusually large or small values (e.g. income data).
• Can be computed for open-ended class intervals, since only the middle position matters.
• Can be located graphically (as the point where the "less than" and "more than" ogives intersect).
• Easy to understand — literally "the middle value."

Limitations of median: …