Median and Mode — Computation for Individual, Discrete and Continuous Series

Median. The median is a positional average — the value of the item that lies exactly in the middle of the series when all observations are arranged in ascending (or descending) order of magnitude, so that exactly half the observations lie below it and half lie above it.

Individual series: first arrange the $N$ values in order.

• If $N$ is odd, the median is the value of the $\left(\dfrac{N+1}{2}\right)^{th}$ item.
• If $N$ is even, the median is the average of the $\left(\dfrac{N}{2}\right)^{th}$ and $\left(\dfrac{N}{2}+1\right)^{th}$ items.

Discrete series: arrange the distinct values in order and build a cumulative frequency (cf) column. Locate the first cf that is equal to or just greater than $N/2$; the value of $X$ corresponding to that cf is the median.

Continuous series: first locate the median class — the class whose cumulative frequency is the first to equal or exceed $N/2$ — then apply:

where $l$ is the lower boundary of the median class, $cf$ is the cumulative frequency of the class before the median class, $f$ is the frequency of the median class itself, and $h$ is the median class's width. (Section 5 showed how the same value can also be read off graphically, from the intersection of the two ogives.)

Mode. The mode is the value that occurs with the greatest frequency in a series — the most typical or 'fashionable' value.

Individual and discrete series: the mode is usually found simply by inspection — the value with the highest frequency. When two or more nearby values have close frequencies and the mode is not obvious by inspection, the grouping method (grouping frequencies in twos and threes in different combinations to see which value is consistently the largest) is used to confirm it.

Continuous series: first locate the modal class — the class with the highest frequency — then apply:

where $l$ is the lower boundary of the modal class, $f_1$ is the frequency of the modal class, $f_0$ is the frequency of the class immediately before it, $f_2$ is the frequency of the class immediately after it, and $h$ is the modal class's width. …