Problem Solving

a certain list of 100 data has an average of 6 and standard deviation d, where d is a positive. Which of the following pairs of data when added to the list must result in a list of 102 data with standard deviation less than d?

-6 and 0
0 and 0
0 and 6
0 and 12
6 and 6


ans E

The standard deviation of a list would come down if the data points are closer to the mean.

The mean of the given list is 6 so the data pair 6 and 6 is closest (amongst the given options) to the mean and thus reduces the SD of the entire list (when added to the list)less than the current SD.

sd = sqrt { [(x-x1)^2 + (x-x2)^2 + .... ]/N}

X - mean
x1 is the frist number , x2 is the second.. etc
N = total numbers

Now our aim is to decrease the SD when 2 new number are added...
N will be increased from 100 to 102

x =6..

So not to increase the sum in numerator.. we can add to numbers as 6,6..
as x-6 = 6-6 =0 in both cases but denominator has increased.. thus the overall SD has decreased....