Question

A list of 100 numbers has the largest value 90 and the smallest value 12.9, with a mean 40, a median 50, and a standard deviation 20. However, you accidentally copied the smallest number “12.9” as “1.29”. Is it possible to determine by how much the mean, the median and the standard deviation change? For each quantity, if so, show your computation. Otherwise, explain why

Answer 1

There are 100 Numbers such that :

Maximum = 90

Minimum = 12.9

Mean ( ) = 40

Median = 50

Std. Deviation ( ) = 20

We accidentally copied smallest number as 1.29 instead of 12.9

Change in Mean :

Mean = X_i / n  

   40 =    X_i / 100        X_i  = Sum of Observations = 4000

New Sum of Observations =    X_i ' = 4000 - 12.9 + 1.29 = 3988.39

New Mean ( ' ) =   ​​​​​​​ X_i ' / 100 = 3988.39 / 100 = 39.8839

Change in Std. Deviation ( ) :

² = ( ​​​​​​​ X_i  ² - n ² ) / ( n - 1 )  

   ( ​​​​​​​ X_i  ² - 100 (40 ² ) ) / 99 = 20 ² = 400

  ​​​​​​​ X_i  ²  = 400 ( 99 ) +  100 (40 ² ) = 199600

New Sum of Square of Observations =  ​​​​​​​ X_{i '} ²  = 199600 - 12.9 ² + 1.29 ² = 199435.2541

Hence , ' ²  =  ( ​​​​​​​ X_{i '} ² - n ' ² ) / 99 = ( 199435.2541 - 100 ( 39.8839 ² ) ) / 99 = 407.7041

   New Std. Deviation = ' = ( 407.7041 ) ^0.5 = 20.1917

Change in Median  :

Median = (( n + 1 ) / 2 )^th Observation = 50.5 ^th Obs. = 50

Since , there is a change in the smallest value i.e , from 12.9 to 1.29 and there is no other value in between the two as 12.9 was the smallest obs. of the original data .

Thus , there would be no effect on the median as it is not impacted by change of max . or min. values

Hence , New Median would also be equal to 50 .