If Q is a set of consecutive integers

[pallaviamahajan]

If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.

(2) The median of set Q is 20.

The procedure for finding the standard deviation for a set is as follows:

1) Find the difference between each term in the set and the mean of the set.

2) Average the squared "differences."

3) Take the square root of that average.

Notice that the standard deviation hinges on step 1: finding the difference between each term in the set and the mean of the set. Once this is done, the remaining steps are just calculations based on these "differences."

Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"

(1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

[x - 10, x - 9, x - 8, x - 7, x - 6, x - 5, x - 4, x - 3, x - 2, x - 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10]

Notice that the difference between the mean (x) and the first term in the set (x - 10) is 10. The difference between the mean (x) and the second term in the set (x - 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!

(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation:

Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2(-3)2 + (-4)2 + (-5)2 + (-6)2(-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences: 770

21 = 36 2

3




The square root of this average is the standard deviation: ≈ 6.06

(2) NOT SUFFICIENT: Since the set is consecutive, we know that the median is equal to the mean. Thus, we know that the mean is 20. However, we do not know how big the set is so we cannot identify the difference between each term and the mean.

Therefore, the correct answer is A.

The answer explanation assumes that the difference between consecutive integers is 1 i.e. x, x+1, x+2 and so on (numbers 1, 2, 3, 4 ...) but isn't list of numbers 2, 4, 6, 8 is consecutive and its standard deviation will be different than it is for 1, 2, 3 and 4 ? May be I'm missing something. Please explain. Thanks.

[ronak_svit]

Answer should be C.
The answer does not assume that the series begins from 1. It can start from any number. We need to check if the series has odd or even number of terms. Since the median is 20, the series shall always have ODD number of terms, we cannot have EVEN number of terms, in which case the median will be x.5 ( as we shall take the average of the two middle numbers ).

Now, when we combine 1 and 2 we know that there are ODD number of terms and 20 is the middle number. So we can easily derive the starting and ending integers:) By doing this we can get the average and hence the standard deviation and so on.

[pallaviamahajan]

I have updated the original post with complete answer explanation from MGMAT. The answer given is A. I didn't understand without knowing how consecutive numbers are evely placed we can determine standard deviation.

[Ben Ku]

The standard deviation is a measure of spread. This means that if data points are close together then the SD is smaller; if the data points are very spread out, then the SD is larger.

Looking at statement (1), Any set of 21 consecutive integers will have the same standard deviation because they are spread out in the same way.

Statement (2) provides us no information about how the data points are distributed, so it's not helpful for us to find the SD.

The answer explanation assumes that the difference between consecutive integers is 1 i.e. x, x+1, x+2 and so on (numbers 1, 2, 3, 4 ...) but isn't list of numbers 2, 4, 6, 8 is consecutive and its standard deviation will be different than it is for 1, 2, 3 and 4 ? May be I'm missing something. Please explain. Thanks.

By definition, consecutive integers mean they are separated by 1. If the problem wanted to refer to consecutive even or consecutive odd integers, it would have stated so.

Hope this helps.

[jeffwey]

What if in (1) it tells you that the the set contains even number of integers, for example- there are 10 numbers in the set, would it still be sufficient? x, x+1, X+2, X+3, and the average is (x+1+x+2)/2...

[jnelson0612]

What if in (1) it tells you that the the set contains even number of integers, for example- there are 10 numbers in the set, would it still be sufficient? x, x+1, X+2, X+3, and the average is (x+1+x+2)/2...

You could still figure out the standard deviation, because I could still determine the mean (x + x+9)/2 as well as the difference between the mean and every other value in the list. That's all we need to know! :-)

[jeffwey]

great thanks Jamie! just want to make sure ;)

[tim]

Let us know if there are any further questions about this one.