GATE | GATE CS 2012 | Question 65

Which of the following assertions are CORRECT?
P: Adding 7 to each entry in a list adds 7 to the mean of the list
Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list
R: Doubling each entry in a list doubles the mean of the list
S: Doubling each entry in a list leaves the standard deviation of the list unchanged
(A) P, Q
(B) Q, R
(C) P, R
(D) R, S


Answer: (C)

Explanation: Mean is average.

Let us consider below example
 2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9 

These eight data points have the mean (average) of 5:
 \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5. 

When we add 7 to all numbers, mean becomes 12 so P is TRUE.

If we double all numbers mean becomes double, so R is also
TRUE.

Standard Deviation is square root of variance.
Variance is sum of squares of differences between all numbers
and means.

Deviation for above example
First, calculate the deviations of each data point from the mean, 
and square the result of each:

 \begin{array}{lll} (2-5)^2 = (-3)^2 = 9 && (5-5)^2 = 0^2 = 0 \\ (4-5)^2 = (-1)^2 = 1 && (5-5)^2 = 0^2 = 0 \\ (4-5)^2 = (-1)^2 = 1 && (7-5)^2 = 2^2 = 4 \\ (4-5)^2 = (-1)^2 = 1 && (9-5)^2 = 4^2 = 16. \\ \end{array} 

variance = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = 4. 

standard deviation = \sqrt{ 4 } = 2 

If we add 7 to all numbers, standard deviation won't change 
as 7 is added to mean also. So Q is FALSE.

If we double all entries, standard deviation also becomes 
double.  So S is false. 

References:
https://en.wikipedia.org/wiki/Standard_deviation
http://staff.argyll.epsb.ca/jreed/math30p/statistics/standardDeviation.htm

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