Bunuel wrote:
A set of 12 test scores has an average of 500 and a standard deviation of 50. Which of the following sets of additional test scores, when combined with the original set of 12 test scores, must result in a combined data set with a standard deviation less than 50?
(A) 6 test scores with average of 450 and standard deviation of 50.
(B) 6 test scores with average of 500 and standard deviation of 25.
(C) 6 test scores with average of 550 and standard deviation of 25.
(D) 12 test scores with average of 450 and standard deviation of 25.
(E) 2 test scores with average of 550 and standard deviation of 50.
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:It is not generally true that all of the terms in a set are within one standard deviation of the mean.
However, standard deviation is a measure of the spread of the terms of a set, so we could represent the original set of scores this way:
The oval spans ±1 standard deviation from the mean, where many of the scores will likely be. This simplification is acceptable as long as we represent all of the other data sets the same way, because all we want to do is compare the
relative effects of the new test scores systematically.
For each of the answer choices, we can now overlay the representative ovals for the new data on top of the oval for the original data.
It is clear that all of the data in the answer choices, when added to the original set, will result in a likely or definite increase in the spread of the data
except for answer choice (B), which definitely concentrates the set of scores closer to the original average of 500. Thus, adding the data in answer choice B will result in a smaller standard deviation than that found in the original data set. The correct answer is B.
We can generalize rules for adding a single term to a set as follows:
* Adding a new term
more than 1 standard deviation from the mean generally
increases the standard deviation of a set.
* Adding a new term
less than 1 standard deviation from the mean generally
decreases the standard deviation of a set.
Note that mathematically this is a slight oversimplification, but for the purpose of adding terms to a set of numbers on the GMAT, you can accept this simplification as true.
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