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

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Math Expert
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A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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16 Jun 2015, 04:04
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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.

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Re: A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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16 Jun 2015, 04:24
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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.

In order to bring down the standard deviation, the average of the additional set needs to be as close as possible to the average of Set A. B has the same average, thus it must decrease the original standard deviation of 50 since B has itself a SD less than 50.
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Re: A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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16 Jun 2015, 08:47
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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.

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

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Re: A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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16 Jun 2015, 09:34
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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.

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,

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Re: A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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22 Jun 2015, 06:42
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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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Re: A set of 12 test scores has an average of 500 and a standard deviation  [#permalink]

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19 Jul 2019, 13:30
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Re: A set of 12 test scores has an average of 500 and a standard deviation   [#permalink] 19 Jul 2019, 13:30
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