Bunuel wrote:
MANHATTAN GMAT OFFICIAL SOLUTION: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.
This entire explanation is mathematically incorrect, and the math involved is miles beyond the scope of the GMAT. If you insert a new element into a set that is precisely equal to the mean, the standard deviation always falls (unless the standard deviation was already zero). If you insert a new element into a set that is different from the mean, you are suddenly in a very complicated situation, mathematically, because you are
changing the mean. All of the distances to the mean in the original set have changed, and you get into some thorny math if you want to work out what happens to the standard deviation in this situation. That is math you'd absolutely never need to understand on the GMAT. But if you do know the math, which is complicated, and you check answer C in this question (inserting six new test scores with a mean of 550 and a standard deviation of 25 into a set of twelve values with a mean of 500 and a standard deviation of 50), you find the standard deviation of the resulting 18-element set is roughly 49.3 (the standard deviation is equal to √[(12*50^2 + 6*25^2 + 12*(500 - new mean)^2 + 6*(550 - new mean)^2)/18]). That's less than 50. So answer C is also a correct answer here, and the question has two right answers -- or it has at least two; I'm not going to check the others. Naturally any conceptual justification that rules out a correct answer can't be explaining concepts correctly, and that's what is happening here, unfortunately.
Even the oval plots the explanation uses to explain standard deviation are, conceptually speaking, a bad way to think about standard deviation, not only because the value of standard deviation is more affected by outliers than by values close to the mean, but because by Chebyshev's Inequality (also not tested on the GMAT) it's possible for a lot of values in a set to be more than one standard deviation from the mean. For example, in a set of 100 values with 49 values equal to 0, two values equal to 50, and 49 values equal to 100, exactly 98% of the values in the set are more than one standard deviation away from the mean.
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