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A certain list of 200 test scores has an average  [#permalink]

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Difficulty:   15% (low)

Question Stats: 71% (00:59) correct 29% (01:30) wrong based on 429 sessions

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A certain list of 200 test scores has an average (arithmetic mean) of 85 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 202 test scores with a standard deviation less than d ?

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(D) 85 and 85
(E) 85 and 90

Intuitively, I can see that the answer is the OA. However, can we be sure that the other choices don't reduce the size of the standar deviation?

Source: www.gmathacks.com

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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
A certain list of 200 test scores has an average (arithmetic mean) of 85 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 202 test scores with a standard deviation less than d ?

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(D) 85 and 85
(E) 85 and 90

Intuitively, I can see that the answer is the OA. However, can we be sure that the other choices don't reduce the size of the standar deviation?

Source: http://www.gmathacks.com

The standard deviation of a set shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

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

Closest to the mean are 85 and 85 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD most.

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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
A certain list of 200 test scores has an average (arithmetic mean) of 85 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 202 test scores with a standard deviation less than d ?

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(D) 85 and 85
(E) 85 and 90

Intuitively, I can see that the answer is the OA. However, can we be sure that the other choices don't reduce the size of the standar deviation?

Source: http://www.gmathacks.com

There is only one correct answer in a PS question. Why would you doubt that? And if you did get another option which reduced the size, what would you do then? Which option will you choose?

They have made the option choices very simple by giving you both the points at the mean itself (so numerator stays same but denominator increases by 2). That is what they want to test. Just don't trouble yourself by bothering about the other options.
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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
VeritasPrepKarishma wrote:
There is only one correct answer in a PS question. Why would you doubt that? And if you did get another option which reduced the size, what would you do then? Which option will you choose?

They have made the option choices very simple by giving you both the points at the mean itself (so numerator stays same but denominator increases by 2). That is what they want to test. Just don't trouble yourself by bothering about the other options.

Thanks Karishma. I ask this question because I am using the method used by many succesful test takers. According to them, you shouldn't evaluate only why the OA is the answer, but also identify why the other choices are wrong. That strategy helpmed a lot, specially in the Verbal section, although I am not so sure whether it is very useful in PS questions. There is a basic difference in Verbal and Quant questions. In Verbal, you are looking for the 'best answer' (Certainly in RC and SC. To a much less extent in CR). In Quant, you are looking for the 'only possible answer'. When you get an answer of 5 m/hr to a TSD question, you do not question why the other options (4 m/hr or 8 m/hr) are not correct.
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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
Bunuel wrote:
The standard deviation of a set shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

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

Closest to the mean are 85 and 85 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD most.

Thank you Bunuel and Karishma, I have an additional question. In this case, the choice is D because the mean still being 85 after adding the two new elements. But if we add two different elements instead of the two 85s (like in the other choices), the mean will change too. So the individual distances between the mean and the elements (the new and the old ones) will change too. Therefore, we couldn't be so sure that the standard deviation has increased or decreased.

What do you think about that? You don't expect that, do you? LOL

GMAT doesn't give you a calculator in Quant so it doesn't expect you to do long calculations. The answer will be logical as in this case. They will not give you options in which you will need to calculate each and find out.
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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
Bunuel wrote:
The standard deviation of a set shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

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

Closest to the mean are 85 and 85 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD most.

Thank you Bunuel and Karishma, I have an additional question. In this case, the choice is D because the mean still being 85 after adding the two new elements. But if we add two different elements instead of the two 85s (like in the other choices), the mean will change too. So the individual distances between the mean and the elements (the new and the old ones) will change too. Therefore, we couldn't be so sure that the standard deviation has increased or decreased.

What do you think about that? You don't expect that, do you? LOL

It's all relative. Look for the numbers closest to the mean. The option that adds minimum to the numerator, will be the only one which will reduce d (since there can be only one answer).

If you look at the rest of the options,
(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(E) 85 and 90

Options (B) and (E) will give the same result so you cannot have both in case option (D) is not there.

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(E) 85 and 86

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Re: A certain list of 200 test scores has an average  [#permalink]

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metallicafan wrote:
A certain list of 200 test scores has an average (arithmetic mean) of 85 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 202 test scores with a standard deviation less than d ?

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(D) 85 and 85
(E) 85 and 90

We know that the average for the given set is 85. Thus, any new value, when added and which is not equal to the average WILL contribute to the S.D. However, if any new value which is added,IS the mean, then the only change will be the increased value (202 in this case from the previous case of 200) in the denominator of the S.D formula. Thus, only option D will ALWAYS keeps the numerator intact and increases the denominator--> Decrease in the overall value for the S.D.
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Re: A certain list of 200 test scores has an average  [#permalink]

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VeritasPrepKarishma wrote:
There is only one correct answer in a PS question. Why would you doubt that? And if you did get another option which reduced the size, what would you do then? Which option will you choose?

They have made the option choices very simple by giving you both the points at the mean itself (so numerator stays same but denominator increases by 2). That is what they want to test. Just don't trouble yourself by bothering about the other options.

Thanks Karishma. I ask this question because I am using the method used by many succesful test takers. According to them, you shouldn't evaluate only why the OA is the answer, but also identify why the other choices are wrong. That strategy helpmed a lot, specially in the Verbal section, although I am not so sure whether it is very useful in PS questions. _________________
"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."

My Integrated Reasoning Logbook / Diary: http://gmatclub.com/forum/my-ir-logbook-diary-133264.html

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Retired Moderator Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
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Re: A certain list of 200 test scores has an average  [#permalink]

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Bunuel wrote:
The standard deviation of a set shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

So when we add numbers, which are far from the mean we are stretching the set making SD bigger and when we add numbers which are close to the mean we are shrinking the set making SD smaller.

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

Closest to the mean are 85 and 85 (actually these numbers equal to the mean) thus adding them will definitely shrink the set, thus decreasing SD most.

Thank you Bunuel and Karishma, I have an additional question. In this case, the choice is D because the mean still being 85 after adding the two new elements. But if we add two different elements instead of the two 85s (like in the other choices), the mean will change too. So the individual distances between the mean and the elements (the new and the old ones) will change too. Therefore, we couldn't be so sure that the standard deviation has increased or decreased.

What do you think about that? You don't expect that, do you? LOL
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Re: A certain list of 200 test scores has an average  [#permalink]

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I think, metallicafan wants to validate his logical reasoning in this case. Therefore, here are my 2 cents on this question.

When I did this question, the first question that came in my mind is - What could be the value, which I was assuming, of Standard Deviation. Then, I rechecked the question I found out is value of std deviation should be positive. i.e. no zero and no negative is allowed.

Assume d to be minimum i.e "1" and see the impact of expected data sets on the mean and std deviation, by using the definition and concept that Bunuel provided.

example - Take choice A . 80 and 80, since you have assumed the d to be 1, therefore, adding these two values will increase the mean and hence standard deviation. Therefore, incorrect.

Now, check the impact of each n every choice.

Thanks
H

metallicafan wrote:
A certain list of 200 test scores has an average (arithmetic mean) of 85 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 202 test scores with a standard deviation less than d ?

(A) 80 and 80
(B) 80 and 85
(C) 80 and 90
(D) 85 and 85
(E) 85 and 90

Intuitively, I can see that the answer is the OA. However, can we be sure that the other choices don't reduce the size of the standar deviation?

Source: http://www.gmathacks.com
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Joined: 02 Sep 2009
Posts: 59625
Re: A certain list of 200 test scores has an average  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: A certain list of 200 test scores has an average  [#permalink]

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Standard deviation is a measurement of how far the values are from the mean

So the smaller the standard deviation it indicates that the values are closer to the value of the mean and the larger the standard deviation, it tells us that the values are farther away from the mean

Keeping this principle in mind, the above set has a mean of 85 and sd = d.

In order to decrease the standard deviation, the two values that we pick must be very close to the mean hence Choice D is the best answer
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Re: A certain list of 200 test scores has an average  [#permalink]

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<--SD---Mean--+SD-->

To reduce SD, pick values which are as close to Mean, so 85 is the answer.

I have a question to experts. If the question asks, whats the minimum value added so that we can have max SD, we shall have two answers 1 on either side, right ?
Also such questions may not have any max value to be added, as max value is infinite. Please correct me if mu understanding is wrong.
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Re: A certain list of 200 test scores has an average  [#permalink]

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ygdrasil24 wrote:
<--SD---Mean--+SD-->

To reduce SD, pick values which are as close to Mean, so 85 is the answer.

I have a question to experts. If the question asks, whats the minimum value added so that we can have max SD, we shall have two answers 1 on either side, right ?
Also such questions may not have any max value to be added, as max value is infinite. Please correct me if mu understanding is wrong.

If you want to increase SD, you will add a value as far away from mean as possible. Say in test scores, if the average is 85 and the maximum marks are 100, the test score that when added will increase SD the most will be 0 (assuming there is no negative marking). 100 is not as far as 0 from the average 85.

And yes, such questions may not have any max value since it may be infinite.
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Re: A certain list of 200 test scores has an average  [#permalink]

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VeritasPrepKarishma wrote:
ygdrasil24 wrote:
<--SD---Mean--+SD-->

To reduce SD, pick values which are as close to Mean, so 85 is the answer.

I have a question to experts. If the question asks, whats the minimum value added so that we can have max SD, we shall have two answers 1 on either side, right ?
Also such questions may not have any max value to be added, as max value is infinite. Please correct me if mu understanding is wrong.

If you want to increase SD, you will add a value as far away from mean as possible. Say in test scores, if the average is 85 and the maximum marks are 100, the test score that when added will increase SD the most will be 0 (assuming there is no negative marking). 100 is not as far as 0 from the average 85.

And yes, such questions may not have any max value since it may be infinite.

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Re: A certain list of 200 test scores has an average  [#permalink]

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