# Average

An average of a list of data is the expression of the central or typical value of a set of data. Mathematically it is defined as the ratio of summation of all the data to the number of units present in the list.

For example:-

1) Find the average of 2, 4 , 6, 8.

Solution:-

Add the numbers = 2 + 4 + 6 + 8 = 20

Total Units = 4

Hence,

$\large&space;Average&space;=&space;\frac{20}{4}&space;=&space;5$

2) Find the average of 6, 13, 17, 21, 23

Solution:-

Add the numbers = 6 + 13 + 17 + 21 = 60

Total Units = 5

Hence,

$\large&space;Average&space;=&space;\frac{60}{5}&space;=&space;12$

By closely analysing these examples, one can observe that the average of a certain list of numbers is the central value of the set. Thus, Average or mean is a quantity intermediate of a set of quantities. In Mathematics, this is also called Average Mean.

## Average of Negative Numbers

If there are negative numbers present in the list, then also the process or formula to find out the average is the same. Let’s understand this with an example.

3) Find the average of 3, −7, 6, 12, −2

Solution:-

The sum of these numbers

= 3 + (-7) + 6 + 12 + (-2)

= 3 – 7 + 6 + 12 – 2

= 12

Total Units = 5

Hence,

$\large&space;Average&space;=&space;\frac{12}{5}&space;=&space;2.4$

How does this whole idea of average or mean works? Average helps you to calculate on how to make all the units present in a list equal.

## Formula for Arithmetic mean

The Arithmetic mean is the most common type of Average. If n numbers are given, each number denoted by ai(where i = 1,2, …, n), the arithmetic mean is the sum of the as divided by n, then

$\large&space;AM&space;=&space;\frac{1}{n}\sum_{i=1}^{n}a_{i}&space;=&space;\frac{1}{n}&space;\;&space;(a_{1}&space;\;&space;+&space;\;&space;a_{2}&space;\;&space;+&space;\;&space;a_{3}&space;\;&space;+&space;\;&space;...&space;\;&space;+&space;\;&space;a_{n})$

where n is the number of observation,

i represent index of summation,

and $a_{i}$ = data value for the given index.

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#### Practise This Question

A car moving on a straight road covers one third of the distance with 20 km/hr and the rest with 60 km/hr. The average speed is