Measures of Central Tendency: Mean, Median and Mode

Arithmetic Mean is the sum of all observations divided by their number. For ungrouped data:

For data given as a discrete frequency series (each value X occurring with frequency f):

Median is the value of the middle-most observation when the data is arranged in ascending (or descending) order — it divides the series exactly into two equal halves. For an ungrouped series with n observations in order, the median is the value of the $\left(\frac{n+1}{2}\right)^{th}$ observation (for odd n). For a continuous (grouped) series, the median lies within the median class — the class in which the cumulative frequency first equals or exceeds $\frac{N}{2}$ — and is computed as:

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.

Mode is the value that occurs most frequently in a series — the most 'typical' or fashionable value. For a continuous series, the mode lies within the modal class — the class with the highest frequency — and is computed as:

where L = lower limit of the modal class, $f_1$ = frequency of the modal class, $f_0$ = frequency of the class preceding it, $f_2$ = frequency of the class succeeding it, and h = class width. …