If the average of ten numbers is 10, then which of the following could

Question

If the average of ten numbers is 10, then which of the following could be the standard deviation of the ten numbers?

I. 0 
II. 10 
III. 20

A. I only 
B. I and II
C. I and III 
D. II and III 
E. I, II, and III

EMPOWERgmatRichC · Answer

Hi All,

Standard Deviation is a relatively rare concept on the GMAT (you'll likely see it just once on Test Day) and while the 'math' behind calculating the Standard Deviation of a group of numbers can be quite 'thick', the GMAT will NEVER ask you to actually perform that calculation. Your basic understanding of the concept is what you'll be tested on.

Here is a hint as to how to think about this particular prompt:

Standard Deviation is really a measure of how 'spread out' a group of numbers is. If there is no spread (meaning the range is 0), then the SD = 0. As the individual numbers become more and more spaced out, the SD rises. Knowing THAT, how spread out could THESE individual numbers get? How high could the SD become?

GMAT assassins aren't born, they're made,
Rich

ziyavutdinov · Answer

How E becomes the answer? Can anyone explain this problem with examples?

chetan2u · Answer

Hi,
the standard deviation measures the spread of the numbers away frm mean, so it can take any value out of teh mentioned three..
example..
SD 0- 10,10,10,10,10
SD 10- 0,0,10,20,20
SD 20- -10,-10,10,30,30
all the above have same mean 10..
hope it helped

EMPOWERgmatRichC · Answer

Hi ziyavutdinov,

The 'key' to this question is that you're NOT supposed to have to come up with specific examples - understanding the concepts involved means that you don't actually have to do that type of work.

We know that we have 10 numbers (and they could be ANY values - including negatives and non-integers) and we know that the average of those numbers is 10. This is meant to say that there are an INFINITE number of possible sets of 10 numbers that fit this prompt.

IF all ten numbers were the same (meaning that we had ten 10s), then the SD would be 0. The more 'spread out' those ten numbers get, the higher the SD becomes. Since there's no limit to the 'spread', then there's no limit to what the SD could be either. Thus, it could be 10 or 20 or any other number. That's why the final answer is E.

GMAT assassins aren't born, they're made,
Rich

BrentGMATPrepNow · Answer

First recognize that, if all 10 numbers are 10 to get the set  {10, 10, 10, 10, 10, 10, 10, 10, 10, 10} , then  the standard deviation of the set is 0 .

Now let's take one of the numbers in the set and change it to  11  to get  {10, 10, 10, 10, 10, 10, 10, 10, 10,  11 } . 
In this case, the  standard deviation is greater than 0

If we change that number to  12  to get  {10, 10, 10, 10, 10, 10, 10, 10, 10,  12 } , then the standard deviation of the set is even bigger. 
In fact, by increasing the value of that last number, we kind of make the standard deviation as big as we want.

In other words, the set of values can have ANY standard deviation from 0 to any positive number.

Answer: E

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JeffTargetTestPrep · Answer

n = 10 and average = 10.

I. If the data set is {10, 10, 10, 10, 10, 10, 10, 10, 10, 10}, then the average is 10, and the standard deviation is 0  .  Checkmark.

II. If the data set is {0, 0, 0, 0, 0, 20, 20, 20, 20, 20}, then the average is 10, and the standard deviation is 10  .  Checkmark.

III. If the data set is {-10, -10, -10, -10, -10, 30, 30, 30, 30, 30}, then the average is 10, and the standard deviation is 20  .  Checkmark.

Answer: E

Note: The standard deviation is the (quadratic) average of the (absolute) deviations from the mean in a data set. So, if the absolute deviations are all the same, then the standard deviation is equal to that same value.

RajibDhara · Answer

if ( 0, 10 and 20 could be the answer ) what could not be the std in this problem, like and infinite number or not greater than some number ?
I got the 0 as a answer but was not sure of 10 and 20

Bunuel · Answer

The standard deviation of a list 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 a high standard deviation indicates that the data are spread out over a large range of values. So, basically we can say that it in a sense measures the distance and the distance cannot be negative, which means that  the standard deviation of any list is greater than or equal to zero :  SD\geq0 . Next, the standard deviation of a list is zero if and only the set consists of identical numbers (or which is the same if the list consists of only one number).

Therefore, the standard deviation of a list having a mean of 10 can be any non-negative value, since the list can be as widespread as we want.

Hope it's clear.

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If the average of ten numbers is 10, then which of the following could

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If the average of ten numbers is 10, then which of the following could

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