A set of 12 test scores has an average of 500 and a standard deviation

No. of Terms in the set = 12
Average = 500
Standard Deviation = 50

CONCEPT: Standard Deviation is defined as Average Deviation of the terms in the set from the Mean value of the Set

For Standard Deviation to become smaller the new set must have terms closer to each other or the new terms must be closer to the mean value of the set so that the average deviation of the terms from the mean value becomes a smaller number.

Observing the 5 option we can see that

Option B has the same Average as the average of the Primary set with 12 terms
also Option B has the lesser Standard Deviation in comparison to the Standard deviation of the Primary set with 12 terms

Hence, It will bring the Standard deviation of the combined set lower than the previous standard deviation and the new combined set of 18 terms will have a standard deviation somewhere between 25 and 50

Answer: Option

Solution -

General Rule: To reduce the Standard Deviation(SD), add more values equal to the mean to the existing list. So that the distribution of values from mean will automatically reduces and results lower SD.

Example - {2,2,2,2} - For this range SD is 0, because the list of values not deviated from mean=2.

In this question, A set of 12 test scores has an average of 500 and a standard deviation of 50.

Add more values equal to average of 500 with lower SD, automatically reduces the SD of list A.

Only ANS B suits

Thanks,

Kudos Please.

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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