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# A certain list of 300 test scores has an arithmetic mean of 75 and a s

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A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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19 Aug 2015, 02:23
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A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Kudos for a correct solution.

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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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19 Aug 2015, 02:28
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Bunuel wrote:
A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Kudos for a correct solution.

If we need to reduce the SD for a set, then we need add the numbers which are closer to the Mean. Since adding a number closer to the mean will shrink the set.

Since the numbers 75 & 75 are equal to the mean, this will reduce the SD.

So the answer is D.
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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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19 Aug 2015, 03:00
Bunuel wrote:
A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Ans: D

Deviation means spreading of the numbers from one median point value..
as the set is already established and we need to add only two numbers, this means we can only add numbers which are closer to the median to reduce the deviation.
so 75 and 75 are the most close numbers and they will increase the number of elements in the set without changing the median value.
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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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23 Aug 2015, 11:45
Bunuel wrote:
A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

As discussed HERE, the standard deviation of a set measures the deviation from the mean. A low standard deviation indicates that the data points are very close to the mean whereas a high standard deviation indicates that the data points are spread far apart from the mean.

When we add numbers that are far from the mean, we are stretching the set and hence, increasing the SD. When we add numbers which are close to the mean, we are shrinking the set and hence, decreasing the SD.

Therefore, adding two numbers which are closest to the mean will shrink the set the most, thus decreasing SD by the greatest amount.

Numbers closest to the mean are 75 and 75 (they are equal to the mean) and thus adding them will decrease SD the most.

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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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23 Dec 2016, 23:47
Great Question.
Here is what i did in this one =>

Using the Basic definition of standard deviation => It is just a statistical tool that is used to measure the average dispersion/variation/deviation/distance of individual data elements with respect to the arithmetic mean => To reduce this we should add data values as close to the mean as possible.

Hence for a compulsory decrease in standard deviation => We should add 75,75(As mean =75)

Hence D

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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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19 Dec 2017, 20:42
To minimize standard deviation, dispersion of the data-points or values should be minimum aroung the mean i.e. selected values should be near the mean.
Best points would be values equal to mean.
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Re: A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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11 Jul 2019, 19:01
Bunuel wrote:
A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Kudos for a correct solution.

The standard deviation is a measure of spread of the data about the mean. Adding data values that are far from the mean make the standard deviation larger. Adding data values equal to the mean makes the spread of the data (i.e., the standard deviation) smaller.

Since the mean is 75, we see that when we add 75 and 75 to the list, the standard deviation must decrease. Note: Depending on the value of d, the other choices might or might not decrease the standard deviation.

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A certain list of 300 test scores has an arithmetic mean of 75 and a s  [#permalink]

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11 Jul 2019, 21:33
Bunuel wrote:
A certain list of 300 test scores has an arithmetic mean of 75 and a standard deviation of d, where d is positive. Which of the following two test scores, when added to the list, must result in a list of 302 test scores with a standard deviation less than d?

(A) 75 and 80
(B) 80 and 85
(C) 70 and 75
(D) 75 and 75
(E) 70 and 80

Kudos for a correct solution.

If 75 & 75 are added to the list, then arithmetic mean will remain the same, since 75 is the mean of 300 test scores.
But Standard deviation will reduce since there is 0 difference between mean and values of these 2 test scores.

New standard deviation will be = $$\sqrt{(300 * d^2 +0 +0)/302}= d * \sqrt{300/302} < d$$

IMO D
A certain list of 300 test scores has an arithmetic mean of 75 and a s   [#permalink] 11 Jul 2019, 21:33
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