## Averages

### 1. Introduction

There will be about $1$ to $3$ questions on Averages in the Quantitative Ability section in entrance tests. Additionally, Data Interpretation questions also typically involve Percentages, Ratios and/or Averages. Therefore, it is imperative for you to have a strong understanding of Averages.

As a student, you might have calculated your own average marks across subjects, or the average marks of your class in a particular subject. Other common examples of averages in real life include characteristics such as height or weight. Averages are used in statistics to showcase the approximate values of any variable, such as the average income of a state, the average rainfall in a year, the average amount spent on food by families etc.

Data points tend to cluster around a central value. This is called central tendency. The commonly used measures of such central tendency are Mean, Median and Mode.

By Mean, we typically refer to Arithmetic Mean or average. The other kinds of Means - Geometric Mean and Harmonic Mean - are covered later in this chapter.

Other statistical concepts such as variance and standard deviation have not come up in entrance tests, and hence, we are not covering these concepts.

### 2. Arithmetic Mean or Average

Arithmetic Mean (AM) or Average is the sum of each item of a data set divided by the total number of items in the data set.

The formula for arithmetic mean, denoted as $\overline{x}$ for $n$ items, namely $x_{1}, x_{2}, x_{3}, ..., x_{n}$ , is

$\overline{x} = \dfrac{x_{1} + x_{2} + x_{3} + ... + x_{n}}{n} \implies \overline{x} = \dfrac{\displaystyle\sum_{i=1}^n x_{i}}{n}$

Note: $\displaystyle\sum_{i=1}^n x_{i}$ is a representation of “sum of all $x_{i}$ values, where $i$ takes all integer values from $1$ till $n$ (both inclusive)”. Therefore, $\textstyle\sum_{i=1}^n x_{i} = x_{1} + x_{2} + x_{3} + ... + x_{n}$

### Example 1

Mr. And Mrs. Roy have 5 children whose weights are 50, 56, 48, 32 and 40 kg. What is the average weight of their children?

### Solution

Average weight $= \dfrac{50 + 56 + 48 + 32 + 40}{5}$

$= \dfrac{226}{5} = 45.2$ kg

Answer: $45.2$ kg