Set X consists of 100 numbers. The average (arithmetic mean)

Question

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

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

Bunuel · Accepted Answer

"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. Answer: E. 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: ps-questions-about-standard-deviation-85897.html ds-questions-about-standard-deviation-85896.html

Bunuel · Answer

"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. Answer: E. Similar questions to practice: https://gmatclub.com/forum/a-certain-lis ... 97473.html https://gmatclub.com/forum/a-certain-lis ... 87743.html https://gmatclub.com/forum/a-certain-lis ... 31448.html https://gmatclub.com/forum/new-ds-set-15 ... l#p1211907 https://gmatclub.com/forum/ps-questions- ... 85897.html https://gmatclub.com/forum/lately-many-q ... 85896.html Hope it helps.

TraderAK · Answer

Answer E
 
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.

TheGmatTutor · Answer

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.

amitjash · Answer

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.

Hussain15 · Answer

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?

Hussain15 · Answer

I think the red portion in my statement above is not correct. SD will come down. :oops:

TheGmatTutor · Answer

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.

ScottTargetTestPrep · Answer

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.

Answer: E

aragonn · Answer

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.

Answer = E

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Set X consists of 100 numbers. The average (arithmetic mean)

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