Average Tips and Tricks for Government Exams

Introduction

The result obtained by adding several quantities together and then dividing this total by the number of quantities is called Average. The main term of average is equal sharing of a value among all, where it may share persons or things. We obtain the average of a number using formula that is sum of observations divide by Number of observations.
Average = Sum of Quantities/Number of quantities ⇒ S = A × N

Theory

Averages can be defined as the central value in a set of data. Average can be calculated simply by dividing the sum of all values in a set by the total number of values. In other words, an average value represents the middle value of a data set. The data set can be of anything like age, money, runs, etc.
Average = [ Sum of Data (or observations) in a set / Number of data (or observations) in that set]

Tips 

• If all the numbers are increased by 'a’, then their average will also be increased by 'a’.
• If all the numbers are decreased by ‘a’, then their average will also be decreased by 'a’.
• If all the numbers are multiplied by 'a’, then their average will also be multiplied by 'a'.
• If all the numbers are divided by 'a', then their average will also be divided by 'a’.

Formula

• Average = Sum of quantities/ Number of quantities
• Sum of quantities = Average * Number of quantities
• The average of first n natural numbers is (n +1) / 2
• The average of the squares of first n natural numbers is (n +1)(2n+1) / 6
• The average of cubes of first n natural numbers is n(n+1)2 / 4
• The average of first n odd numbers is given by (last odd number +1) / 2
• The average of first n even numbers is given by (last even number + 2) / 2
• The average of squares of first n consecutive even numbers is 2(n+1)(2n+1) / 3
• The average of squares of consecutive even numbers till n is (n+1)(n+2) / 3
• The average of squares of squares of consecutive odd numbers till n is n(n+2) / 3
• If the average of n consecutive numbers is m, then the difference between the smallest and the largest number is 2(m-1)
• If the number of quantities in two groups be n₁ and n₂ and their average is x and y respectively, the combined average is (n₁x + n₂y) / (n₁ +  n₂)
• The average of n quantities is equal to x. When a quantity is removed, the average becomes y. The value of the removed quantity is n(x-y) + y
• The average of n quantities is equal to x. When a quantity is added, the average becomes y. The value of the new quantity is n(y-x) + y

Solved Examples