Data Sufficiency

List X contains 8 integers. Is the standard deviation of the integers in list X equal to zero?

(1) The range of the integers in list X is 3.

(2) The average (arithmetic mean) of the integers in list X is 5.


OA A

j_shreyans wrote:List X contains 8 integers. Is the standard deviation of the integers in list X equal to zero?

(1) The range of the integers in list X is 3.

(2) The average (arithmetic mean) of the integers in list X is 5.


OA A

One simple way to see standard deviation is as the average of the distances between the values in a list and the mean of the values. So basically, if not exactly, standard deviation is average deviation from the mean.

What this means is that the only way that the standard deviation of a list of values can be zero is by all the values in the list being the same as the mean. If any are different from the mean, then there is some deviation from the mean, and so the average deviation will not be zero.

Statement 1 tells us that there is a difference between the highest and lowest values in the list. If there is a difference between at least two of them, they can't all equal the mean. So there is some deviation and the standard deviation cannot be 0.

So Statement 1 is sufficient.

Statement 2 merely gives us the mean without providing information regarding the values that underlie that mean. So there is no way to determine anything about the standard deviation from the information in Statement 2.

So the correct answer is A.

j_shreyans wrote:List X contains 8 integers. Is the standard deviation of the integers in list X equal to zero?

(1) The range of the integers in list X is 3.

(2) The average (arithmetic mean) of the integers in list X is 5.

SD=0 if NONE of the values DEVIATES from the mean.
Put another way:
SD=0 if ALL OF THE VALUES ARE THE SAME.
Question stem, rephrased:
Are all of the values the same?

Statement 1:
biggest - smallest = 3.
biggest = smallest + 3.
Since the biggest value is 3 greater than the smallest value, all of the values are NOT the same.
SUFFICIENT.

Statement 2:
Sum of the 8 integers = (number)(average) = 8*3 = 24.

Case 1: {3, 3, 3, 3, 3, 3, 3, 3}
In this case, all of the values are the same.

Case 2: {3, 3, 3, 3, 3, 3, 2, 4}
In this case, all of the values are NOT the same.
INSUFFICIENT.

The correct answer is A.

Since standard deviation measures the dispersion in your set -- that is, how spread out the numbers are -- it can only be 0 if there's NO dispersion: if the numbers are NOT spread out at all.

S1 tells us that the numbers ARE spread out: they have a range, so at least two of them are different. That means the standard deviation CANNOT be 0: there is dispersion, so the measure of it is positive.

S2 doesn't tell us anything: the set could be {5, 5, 5, 5, 5, 5, 5, 5}, in which case the s.d. is 0, or it could be anything else, in which case the s.d. > 0.