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Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

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Originally posted by Hussain15 on 30 May 2010, 06:49.
Last edited by Bunuel on 24 Sep 2015, 03:52, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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4
4
Hussain15 wrote:
Hussain15 wrote:
I have another doubt:

If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum.

What am I missing here?

I think the red portion in my statement above is not correct. SD will come down. "Standard deviation shows how much variation there is from the mean. 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 10 and 10 (actually these numbers equal to the mean) thus adding them will shrink the set most, thus decreasing SD by the greatest amount.

amitjash wrote:
I have a doubt for this explaination.
The question says "will decrease the set’s standard deviation by the GREATEST amount" Now if SD is +ve then adding a negetive number will reduce the standard deviation correct?? and if SD is -ve then adding the positive number will reduce the standard deviation. the value of the added will depend on the number of data given.
Please correct me if i am wrong.

SD is always $$\geq{0}$$. SD is 0 only when the list contains all identical elements (or which is same only 1 element).

For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice:
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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Hussain15 wrote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 & -100
B. -10 & -10
C. 0 & 0
D. 0 & 20
E. 10 & 10

Standard Deviation is deviation from the mean. If all the numbers in a set are equal to mean, then the standard deviation will be zero.
Therefore, in the given set with mean of 10, if we add 10 & 10, then the standard deviation will reduce. All other numbers will increase the standard deviation.

Please note that B is not the answer. Mean is +10 and, therefore, -10 is 20 points away from +10 (its not equal to the mean). Therefore, -10 will increase the standard deviation of the given set.
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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Clearly E, as the mean of the set is 10. Adding the mean as an extra value in the set will decrease the standard deviation.
Manager  Joined: 17 Mar 2010
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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I have a doubt for this explaination.
The question says "will decrease the set’s standard deviation by the GREATEST amount" Now if SD is +ve then adding a negetive number will reduce the standard deviation correct?? and if SD is -ve then adding the positive number will reduce the standard deviation. the value of the added will depend on the number of data given.
Please correct me if i am wrong.
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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I have another doubt:

If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum.

What am I missing here?
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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Hussain15 wrote:
I have another doubt:

If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum.

What am I missing here?

I think the red portion in my statement above is not correct. SD will come down. _________________
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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Yes, this part is incorrect.

There will be no change in the average if you add (10, 10), but the difference will be zero, when we calculate for SD. So, SD will be <4.6.

E is correct, which brings the miminum sum after subtracting and squaring the differences for SD calculations.

Hussain15 wrote:
Hussain15 wrote:
I have another doubt:

If we will add 10 & 10 to set X, then the arithematic mean will not change i.e it will still be 10. Hence the standar deviation will not change & will remain the same. But our requirement is that SD will decrease in maximum.

What am I missing here?

I think the red portion in my statement above is not correct. SD will come down. Manager  Joined: 04 Apr 2010
Posts: 77
Schools: UCLA Anderson
Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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2
You can prove it yourself with a couple quick calculations. Try calculating the SD for the set

{1,2,3}

and then calculate it for the set

{1,2,2,3}

Notice how it decreases.
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shrive555 wrote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

"Standard deviation shows how much variation there is from the mean. 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 10 and 10 (actually these numbers equal to the mean) thus adding them will shrink the set most, thus decreasing SD by the greatest amount.

For more on this issue please check Standard Deviation chapter of Math Book (link in my signature) and the following two topics for practice:

Hope it helps.
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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Bunuel wrote:
gmatgambler wrote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A)-100 and -100
B)-10 and -10
C)0 and 0
D)0 and 20
E)10 and 10

Merging similar topics. Please refer to the solutions above.

Theory on Statistics and Sets problems: math-standard-deviation-87905.html

All DS Statistics and Sets problems to practice: search.php?search_id=tag&tag_id=34
All PS Statistics and Sets problems to practice: search.php?search_id=tag&tag_id=55

Similar questions to practice:
a-certain-list-has-an-average-of-6-and-a-standard-deviation-97473.html
a-certain-list-of-100-data-has-an-average-arithmetic-mean-87743.html
a-certain-list-of-200-test-scores-has-an-average-131448.html
new-ds-set-150653-60.html#p1211907

Hope this helps.
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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How I think about problems associated with means and standard deviations is with scattered data points. Since this question is regarding standard deviations (and how to reduce it), a quantity calculated to indicate the extent of deviation for the group -- the greater the deviation from the mean the greater the standard deviation. Therefore what data points will help to reduce deviation from the mean of set X . If you add two points with the value of the mean their deviation will be 0, which is the smallest deviation you can add to the set of numbers.
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Re: Set X consists of 100 numbers. The average (arithmetic mean) of set X  [#permalink]

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i reviewed math of Gmat club and found that 10,10 will not decrease the SD
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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HIgh SD = Wide distribution of data in the set
Low SD = Closely associated data in the set

The intention is to reduce the SD

That implies the newer entrants into the set need to near the mean

Option A would Increase the SD the most
Option E would not affect as much.

Hence E
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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Hussain15 wrote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

The standard deviation is a measure of the spread of data values around the mean. If data values are close to the mean, the standard deviation is small, and if data values are further away from the mean, the standard deviation is larger.

Thus, in order to decrease the standard deviation, we want to find two values that are as close to the mean as possible. Since the mean is 10, the two values of 10 and 10 will decrease the standard deviation by the greatest amount.

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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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niks18 Bunuel chetan2u

Quote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

If one of the answer option is -10 and 10, will it mean same as (C)?
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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niks18 Bunuel chetan2u

Quote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

If one of the answer option is -10 and 10, will it mean same as (C)?

Can you elaborate your query? If Option is not present and instead of Option E, your option of -10 and 10 is given then in my opinion none of the answer choice will decease the SD but will increase the SD. So the question will become irrelevant.

Between Option C i.e. 0 & 0 and your option -10 & 10, 0 & 0 will have a less incremental impact on SD than -10 & 10 because both 0 & 0 and -10 & 10 will only reduce the average of the set but when you subtract each element of the set with the reduced average and then square it to get the variance, then -10 & 10 will give you higher variance than 0 & 0, resulting in higher SD
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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Official Explanation:

Standard deviation is a tricky subject. Here’s a GMAT blog that explains absolutely everything you need to know about the standard deviation.

First we have to understand what the mean is. The mean is just the plain old average --- add up all the terms on the list, and divide by the number of terms on the list. Adding -100 and -100 would lower the mean the most, but that’s not what the question is asking.

The standard deviation is something much more complicated. There's a technical definition, which I explain in that GMAT blog, but the more informal definition is fine for this problem.

The informal definition --- standard deviation is sort of an average of the deviations from the mean. You see, every term has its own distance from the mean, and we call that a “deviation from the mean” ---- for these purposes, we count it as positive whether it’s greater than or less than the mean.

In this problem, the mean = 10. A value of 13 would have a deviation from the mean of 3. A value of 7 would also have a deviation from the mean of 3. Each of those terms, 13 and 7, are a distance of 3 from the mean, so in terms of the standard deviation, they contribute the same thing. Very high or very low numbers, far away from the mean, would have large deviations from the mean. We find the deviation from the mean of every number on the list --- all 100 numbers in this problem ---- and the standard deviation is sort of an average of all 100 of those deviations from the mean. (Again, this is not the precise definition, it’s not a strict average, but for this problem it’s close enough.)

The question gives a numerical value for the standard deviation, 4.6, but that's just a distractor. We don't need that.

The question asks us to add two numbers that lower the standard deviation the most. Well, the standard deviation is a kind of average, and if we want to lower any average, we have to add new terms that make contributions as small as possible. What's the smallest possible deviation a number could have from the mean? If we added a new term with a value of 10, that term equals the mean, so its deviation from the mean—its distance from the mean—is zero. If the new term is anything above or below 10, it will have a deviation from the mean greater than zero. Therefore, the lowest possible deviation from the mean a term can have is zero, and this is possible only if the term equals the mean of 10. Thus, if we two terms, both equal to the mean, we will be adding two terms with the lowest possible deviation from the mean, and that will lower the standard deviation, the average across all deviations from the mean, as much as possible.

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Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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Hussain15 wrote:
Set X consists of 100 numbers. The average (arithmetic mean) of set X is 10, and the standard deviation is 4.6. Which of the following two numbers, when added to set X, will decrease the set’s standard deviation by the greatest amount?

A. -100 and -100
B. -10 and -10
C. 0 and 0
D. 0 and 20
E. 10 and 10

if the question had asked "...will decrease the set’s standard deviation by the LEAST amount .
the answer would be (0,20) & (0,0) & (-10,10) right?
cause : -100, -100 will make numerator very large , thus increasing the SD.
only (0,0) & (0,20) & (-10,-10) will add 100,100 to the numerator ( least among all)

for increase by greatest , answer would be -100,-100
for increase by least , answer = cant say
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Re: Set X consists of 100 numbers. The average (arithmetic mean)  [#permalink]

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_________________ Re: Set X consists of 100 numbers. The average (arithmetic mean)   [#permalink] 31 Dec 2019, 13:05
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