Median of Percentages

The median is the value that splits an ordered data set in half: at least 50% of the values are at or below it, and at least 50% at or above.

Percentages are just numbers on a $0$–$100$ scale, so the procedure is the ordinary one — but only if every percentage is measured on the same kind of base. Six departments' pass rates can be sorted and a median taken. A 90% rate from 10 students and a 40% rate from 4,000 students are still one data point each; the median deliberately ignores how big each group is.

The critical caveat: the median of subgroup percentages is not the overall percentage. To get the overall rate, add the numerators and add the denominators. The median answers "what is a typical department?", not "what fraction of all students passed?"

Why use it at all? The median is resistant. Rates bounded at 0 and 100 are often skewed, and one 0% or 100% group would drag a mean but leave the median untouched.

1. Sort all values in ascending order. Do not skip this — an unsorted list is the most common source of a wrong answer.
2. Count the values, $n$, including repeats.
3. Locate position $\dfrac{n+1}{2}$.

If $n$ is odd, that position is a whole number and the median is the value sitting there:

$\text{median} = x_{(n+1)/2}$

If $n$ is even, the position lands halfway between two values, so average them:

$\text{median} = \frac{x_{n/2} + x_{n/2 + 1}}{2}$

Grouped data. When percentages are reported in classes, interpolate within the class containing the $n/2$-th observation:

$\text{median} = L + \left(\frac{n/2 - F}{f}\right)h$

where $L$ is the lower boundary of the median class, $F$ the cumulative frequency before it, $f$ its frequency, and $h$ its width. This assumes values are spread evenly inside the class.

• Taking the middle of the list as typed. Always sort first. The middle of an unsorted list is meaningless.
• Reporting the position instead of the value. $(n+1)/2 = 3$ means "the third value", not "3%".
• Using $n/2$ as the position for odd $n$. For $n = 5$ that gives 2.5, not the third value; the formula is $(n+1)/2$.
• Averaging subgroup percentages to get an overall rate. Neither the mean nor the median of rates recovers the pooled rate — this is Simpson's paradox territory.
• Mixing formats. Convert everything to one form ($0.45$ or $45\%$) before sorting; $0.45$ sorts below $8$ if you leave them mixed.
• Dropping duplicates. Repeated values each occupy a position and shift the middle.