Measures of Central Tendency — Mean, Median and Mode

Business decisions are constantly made from data — average sales, typical wages, the most common size of order — and the first job of statistics is to reduce a whole mass of raw numbers to a single representative figure. A measure of central tendency is exactly such a figure: a single value that best represents the entire data set.

Arithmetic Mean. For $n$ ungrouped observations $x_1,x_2,\ldots,x_n$: $\bar{x} = \dfrac{\sum x_i}{n}$. For data grouped into classes with frequencies $f_i$ and mid-points $x_i$: $\bar{x} = \dfrac{\sum f_ix_i}{\sum f_i}$. The mean uses every single value in the data set, which makes it the most commonly used average, but also the one most distorted by an extreme value (an outlier).

Median. The middle value of the data when arranged in ascending (or descending) order — for ungrouped data with $n$ observations, the median is the $\left(\dfrac{n+1}{2}\right)^{\text{th}}$ value if $n$ is odd, or the average of the $\left(\dfrac{n}{2}\right)^{\text{th}}$ and $\left(\dfrac{n}{2}+1\right)^{\text{th}}$ values if $n$ is even. Unlike the mean, the median is not affected by how extreme the highest or lowest values are, only by how many observations lie above and below it — this makes it the preferred average for skewed data, such as income or wealth figures with a few very large values.

Mode. The value that occurs most frequently in the data set. A data set can have one mode (unimodal), two (bimodal), or more; it is the only one of the three averages that can meaningfully describe purely categorical/qualitative data (e.g. 'the most commonly purchased size').

Computing and choosing between mean, median and mode is a universal first step of descriptive statistics, identical in method across every Indian commerce board's statistics syllabus.