Question

The mean of 10 numbers is 24. If one number is included, the new mean is 25. Find the included number. 

Answer 1

Let `barx` be the mean of n number of observation x₁, x_2, x₃,..., x_n 

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

Given that mean of  10 numbers is  24. 
That is,
`(x_1 + x_2 + x_3 +...+ x_10)/(10)`= 24
⇒ x₁ + x₂ + x₃ + ... + x₁₀ = 10 x 24 
⇒ x₁ + x₂ + x₃ + ... + x₁₀ = 240
⇒ x₁ + x₂ + x₃ + ... + x₁₀ + x₁₁ = 240 + x₁₁  ....(1) 

Also, given that mean of 11 number is 25.
That is,

`(x_1 + x_2 + x_3 +...+ x_10 + x_11)/(11) = 25`
⇒ x₁ + x₂ + x₃ + ... + x₁₀ + x₁₁ = 11 x 25
⇒ x₁ + x₂ + x₃ + ... + x₁₀ + x₁₁ = 275        ....( 2 )

From equations ( 1 ) and ( 2 ), we have :
x₁ + x₂ + x₃ + ... + x₁₀ + x₁₁ = 240 + x₁₁ = 275
240 + x₁₁ = 275
⇒ x₁₁ = 275 - 240 = 35