GMAT PREP PS Problem

alex.gellatly 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 o
C. 0 and 6
D. 0 and 12
E. 6 and 6

Thanks

Answer is E.

By definition Standard Deviation S.D. is measured by how far is a value vis-a-vis mean. The lesser the Absolute Difference of values from mean, the lesser is S.D. value and visa-versa.

In present case we wish to get S.D. 'd' lesser than 'd' by adding 2 new values. To get this, we should make Absolute Difference of 2 new values from mean as zero. It is only possible with values 6 & 6 only as |6-6|=0.

Answer is E.

alex.gellatly 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 o
C. 0 and 6
D. 0 and 12
E. 6 and 6

Thanks

One alternative method of solving this is...

As SD = sqrt [ { (X1-mean)^2 + (X2-mean)^2 +.....(Xn-mean)^2) }/ n];

In the present case n = 100 items, mean =6.

So, SD = d= sqrt [ { (X1-6)^2 + (X2-6)^2 +.....(Xn-6)^2) } / 100];

say 2 new values added are a & b, then

Revised SD = sqrt [ { (X1-6)^2 + (X2-6)^2 +.....(Xn-6)^2+ (a-6)^2 + (b-6)^2 ) } / 102 ];

Since we want Revised SD < d, hence we need to make (a-6)^2 + (b-6)^2 minimum. It is possible if (a-6)^2 + (b-6)^2 = 0 as it will attain 0 or +ive values only or a = b = 6 each.

alex.gellatly 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 o
C. 0 and 6
D. 0 and 12
E. 6 and 6

We should keep in mind that the closer the data points are to the mean, the “lower” the standard deviation. Thus, since the mean is 6, when we add two additional values of 6 to the data set, the standard deviation will decrease.

Answer: E