GMAT Question of the Day - Data Sufficiency - Statistics/Standard Deviation
A factory received 19 boxes of widgets. What was the standard deviation of the numbers of widgets in the 19 boxes?
(1) For the 19 boxes of widgets, the median of the numbers of widgets was equal to the mean of the number of widgets.
(2) For the 19 boxes of widgets, the value of the range was less than or equal to the minimum number of widgets.
GMAT Question of the Day Solution
On the GMAT you don't see standard deviation questions very often. However, for most GMAT tutoring students the topic of standard deviation is a bit of a mystery. The most common question: Do I need to know the formula for standard deviation for the GMAT? Nope. I don't know it and have never used it. I've also never seen a GMAT question that requires knowing the formula for standard deviation. Is it possible that there is a GMAT question out there that requires knowing the standard deviation formula? Maybe but I doubt it. That said you do need to understand the concept of standard deviation and what information you need to calculate it.
So what is standard deviation? It boils down to how spread out a set of numbers is. The set 1, 2, 3 has a smaller standard deviation than the set 1, 1000, 5000. How is standard deviation tested on the GMAT? Most often you'll see standard deviation in the data sufficiency. Most often the question will ask whether you can calculate the standard deviation for a certain set of numbers. There are some special GMAT standard deviation rules that you should memorize. These are situations in which you do not need to know the specific numbers in the set but you will be able to calculate the standard deviation.
If all of the numbers in the set are the same then the standard deviation is zero. Here are some clues that indicate that all of the numbers in a set are the same:
1. The range of the set is zero
2. The max of the set equals the average
3. The min of the set equals the average.
If the set is evenly spaced and you know the spacing and the number of numbers then you can calculate the standard deviation. Why? Well, any evenly spaced set with the same spacing and the same number of numbers has the same standard deviation. The standard deviation of 1, 2, 3 is the same as the standard deviation of 45, 46, 47.
In this GMAT question of the day we aren't quite so lucky as to have a cut and dry case but the information above will help a bit.
(1) The median equal to the mean indicates that we have an evenly spaced set (which could also be a set containing only one value, a spacing of 0). However we do not know the number of numbers in the set so we can't calculate the standard deviation.
(2) The range being less than the minimum doesn't provide us with enough information to solve for the standard deviation. You can give yourself some examples that would obviously create different standard deviations. You could have a set of equal numbers. All 7's for instance. The range is 0, the min is 7, and the standard deviation is 0. You could also have a set with a bunch of 15's and a bunch of 16's. In this case the min is 15, the range is 1, and the standard deviation is not 0. Two different values for standard deviation. Insufficient.
(1) + (2) Putting the statements together we have an evenly spaced set with a range that is less than the minimum. You can either have a set of all the same number or a consecutive set that starts at 19 or above (there are other options but you only need two different ones to prove insufficiency). Insufficient. Let's write out a few examples:
7, 7, 7...
101, 102, 103...
A set made of all 7's will certainly have a different standard deviation from a set made up of 100, 101, 102...119