Question

The mean of 6 numbers is 42. If one number is excluded, the mean of the remaining number is 45. Find the excluded number.

Answer 1

Let `barx` be the mean of n number of observation x₁, x₂, x₃, ..., x_n  

Mean of given data = `(x_1 + x_2 + x_3 + ... + x_n)/ (n)`

Given that mean of 6 number is 42.
That is,

`(x_1 + x_2 + x_3 + ... + x_6)/ (6) = 42`

⇒ x₁ + x₂ + x₃ + ... + x₆ = 6 x 42 

⇒ x₁ + x₂ + x₃ + x₄ + x₅ = 252 - x₆   ...( 1 )

Also, given that the mean of 5 number is 45.
That is,

` ( x_1 + x_2 + x_3 + x_4 + x_5)/( 5 ) = 45`

⇒ x₁ + x₂ + x₃ + x₄ + x₅ = 5 x 45

⇒ x₁ + x₂ + x₃ + x₄ + x₅ = 225      ....( 2 )

From equation ( 1 ) and ( 2 ), we have
 x₁ + x₂ + x₃ + x₄ + x₅ = 252 - x₆ = x₁ + x₂ +x₃ + x₄ + x₅ = 225
252 - x₆ = 225
⇒ x₆ = 252 - 225 = 27