Gmatter111 wrote:
A certain list of 100 data has an average (arithmetic mean) of 6 and a standard deviation of d, where d is 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 ?
A. -6 and 0
B. 0 and 0
C. 0 and 6
D. 0 and 12
E. 6 and 6
Logically, SD of a set will decrease if you add numbers which are equal to its mean. Thus the answer should be E.
I will however provide you with mathematical reasoning to justify the above statement.
S.D. for hundred numbers = d = \(\sqrt{\frac{S}{100}}\) where 'S' is sum of the squares of the difference between each number and the mean.
Now let the two numbers added be 'x' and 'y'.
S.D after adding the two numbers will be = \(\sqrt{\frac{S}{102}+\frac{(x-6)^2 + (y-6)^2}{102}}\)
Now it is obvious that \(\frac{S}{102}\) will be less than \(\frac{S}{100}\). Also, the minimum value of \(\frac{(x-6)^2 + (y-6)^2}{102}\) will be 0 when both 'x' and 'y' are equal to 6.
Thus if the two numbers added are equal to the mean, the SD of the set must decrease. (Unless of course SD of the set was 0 to start with (not in our case) and then in that case SD will remain constant).
Answer : E
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