What is Average ?

What is Average ?


The average of a set of numbers is the ratio of the sum of the numbers to the total count of numbers.


The average is the central or middle value of a group of numbers. It shows what each value would be if the total were distributed equally among all items.


In simple words, the average is the equal share for everyone.


Formula: Average  = Sum of all numbers / Total numbers


Ex: Numbers: 2, 4, 6, 8


Soln


Sum = 2 + 4 + 6 + 8 = 20


Count = 4


Average  = 20 ÷ 4 = 5


Special Cases


Average of first n natural numbers: Average = n+1/2


Explanation: Sum of first n natural numbers = n(n+1)/2 → divide by n


Ex: First 10 natural numbers = 1, 2, 3, …, 10


Soln


Average = (10 + 1)/2 = 11/2 = 5.5



  1. Average of first n odd numbers: Average = n


Explanation: First n odd numbers = 1, 3, 5, … → Average  = n


Ex: First 5 odd numbers = 1, 3, 5, 7, 9


Average = 5



  1. Average of first n even numbers: Average  = n+1


Explanation: First n even numbers = 2, 4, 6, … → Average  = n + 1


Ex: First 6 even numbers = 2, 4, 6, 8, 10, 12


Average = 6 + 1 = 7



  1. Average of squares of first n natural numbers: Average  = (n+1)(2n+1) / 6


Explanation: Sum of squares = 12+22+32+…+n2 = n(n+1)(2n+1) / 6​ → divide by n


Ex:


First 5 natural numbers = 1, 2, 3, 4, 5


Squares = 1, 4, 9, 16, 25 → sum = 55


Average = 55 ÷ 5 = 11


Formula check: ((5 + 1)(2*5 + 1))/6 = (6×11)/6 = 11


Importance of Average


The average helps us understand the overall level or performance of a group.
It is widely used in mathematics and daily life - for finding



  • average time,

  • speed,

  • marks,

  • temperature, etc.