Introduction

A large mass of raw data — the marks of every student in a class, the rainfall recorded across an area, the output of a factory month after month, or the income of every family in a locality — is difficult to grasp at a glance. A measure of central tendency is a numerical method that condenses such a set of figures into a single representative value, so the entire data set can be described in brief. We rely on this idea constantly in everyday life: average marks obtained in a test, average rainfall, average production, and average income are all examples of summarising many observations by one number.

The chapter motivates the idea through Baiju, a small farmer who grows food grains in Balapur village (Buxar district, Bihar), a village of 50 small farmers. Baiju owns 1 acre of land, and we want to judge his economic condition relative to the other farmers. Listing all 50 land-holdings and comparing them one by one is confusing; instead we summarise the whole set of holdings in a single value that can represent the entire data. Having done so, we can ask whether Baiju's holding is:

• above average in the ordinary sense — captured by the Arithmetic Mean;
• above the size that half the farmers own — captured by the Median;
• above what most of the farmers own — captured by the Mode.

So the measuring of central tendency is a way of summarising the data in the form of a typical or representative value around which the observations tend to cluster.

There are several statistical measures of central tendency, commonly called "averages". The three most widely used are:

• Arithmetic Mean — the sum of all the values divided by the number of observations, usually denoted $\bar{X}$. For a set of $N$ observations $X_1, X_2, \ldots, X_N$, it is $\bar{X} = \dfrac{\sum X}{N}$. For instance, the mean of six monthly family incomes 1600, 1500, 1400, 1525, 1625 and 1630 is about Rs 1,547, meaning that on an average a family earns Rs 1,547.
• Median — the middle value of the data when arranged in order, so that half the observations lie below it and half above it.
• Mode — the value that occurs most frequently, i.e. the most typical value of the series.

Besides these three, there are two further averages — the Geometric Mean and the Harmonic Mean — which are suitable in certain special situations, but the present discussion is limited to the three averages above.

Studying this chapter enables you to understand why a set of data needs to be summarised by one single number, to recognise and distinguish between the different types of averages, to learn how to compute each of them, to draw meaningful conclusions from a set of data, and — importantly — to judge which average is the most useful in a given situation, since the right choice depends on the purpose of the analysis and the nature of the distribution.