Standard deviation s?

Question

The mean score of 500 students on a statistics test is 45 and s (sample standard deviation) is 5.

If two points were added to each of the 500 scores, the new s would be:
a) 2.5
b)3
c)5
d)10

The answer is c but I'm not sure how?
Would a confidence interval be used in this case?

Same with If two points were subtracted from each of the 500 scores, the new s would also be 5.. (not sure how this works..)

Alfredo Kraus · Accepted Answer

The mean, m is equal to
m = (1/500) Sum(i=1 to 500) xi where the xi are the individual scores.  Clearly, if we add 2 to every x1 the new mean is  (1/500) Sum(i=1 to 500) (xi+2) = m + 2*500/500 = m+2
The square of the sample standard deviation is, originally
s^2 = (1/500) Sum(i=1 to 500) (xi - m)^2
If we add 2 to each xi (and 2 to m as we saw above) we have that the square of the new sample standard deviation is
(1/500) Sum(i=1 to 500) [ (xi +2) - (m+2) ]^2 = (1/500) Sum(i=1 to 500) (xi - m)^2 = s^2 so the new sample standard deviation is exactly the same as the old one so the answer is (c).  Clearly it would still be the same if we had used -2 instead of +2 or indeed if we had used any constant

Mr. Purple Pants · Answer

If I had to guess, you're over-thinking the situation. This is actually a very simple question, so maybe you just need to look back. Now if every single score got an additional two points, that would move the entire sample up by two points. In doing so, the mean would change, but the deviations from that mean wouldn't. That means every single value would remain the same distance from the mean as they were before adding two points, so the standard deviation would not be influenced.

Sorry if this explanation was too long. Just tried to clear up the situation.

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Standard deviation s?

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