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# Decrease in Standard Deviation

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Decrease in Standard Deviation [#permalink]  30 May 2010, 05:49
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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 & -100
B. -10 & -10
C. 0 & 0
D. 0 & 20
E. 10 & 10
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Re: Decrease in Standard Deviation [#permalink]  30 May 2010, 06:22
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: Decrease in Standard Deviation [#permalink]  02 Jun 2010, 16:48
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.
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Re: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 02:41
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: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 05:04
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: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 05:14
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: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 05:43
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.

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Re: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 05:49
Expert's post
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: Decrease in Standard Deviation [#permalink]  04 Jun 2010, 09:48
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.
Re: Decrease in Standard Deviation   [#permalink] 04 Jun 2010, 09:48
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