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There is a set of consecutive even integers. What is the 10 Aug 2010, 08:20
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There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set.
(2) the mean of the set is 382.
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Bunuel
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There is a set of consecutive even integers. What is the 10 Aug 2010, 09:26
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Q: There is a set of consecutive even integers. What is the standard deviation of the set?
(1) There are 39 elements in the set.
(2) the mean of the set is 382.

Answer is a. Can Someone explain

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Two very important properties of standard deviation:

If we add or subtract a constant to each term in a set:
Mean will increase or decrease by the same constant.
SD will not change.

If we increase or decrease each term in a set by the same percent (multiply all terms by the constant):
Mean will increase or decrease by the same percent.
SD will increase or decrease by the same percent.

You can try it yourself:
SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set.

That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant.

Back to the original question:

There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set --> SD of a set of ANY 39 consecutive even integers will be the same, as any set of 39 consecutive even integers can be obtained by adding constant to another set of 39 consecutive integers. For example: set of 39 consecutive integers {4, 6, 8, ..., 80} can be obtained by adding 4 to each term of another set of 39 consecutive integers: {0, 2, 4, ..., 76}. So we can calculate SD of {0, 2, 4, ..., 76} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient.

(2) The mean of the set is 382 --> knowing mean gives us nothing, we must know the number of terms in the set, as SD of {380, 382, 384} is different from SD of {378, 380, 382, 384, 386}. Not sufficient.

Answer: A.

Hope it's clear.
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There is a set of consecutive even integers. What is the 28 Mar 2011, 19:35
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There is a set of consecutive even integers. What is the standard deviation of the set?

(1) There are 39 elements in the set.
(2) the mean of the set is 382.
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There is a set of consecutive even integers. What is the 28 Mar 2011, 20:27
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Pardon me. Let me try. The sd is a measure of the compactness. The consecutive integers are packed the same way ( equidistant ) from each other no matter where you start counting from. To get the distances from the mean of a data sample you just need the number of elements. Let's say it is 5 and data set is {1 2 3 4 5} the mean is middle number 3. Because of symmetry and knowing data is equidistant from each other, we can calculate the square of individual distances from the mean. Hence we can know the variance. Hence we can get the sd. So here A is sufficient to answer the question. because of symmetry the sd of {1 2 3 4 5} is same as the sd of {96 97 98 99 100} or {11 12 13 14 15} ie any consecutive 5 integers

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There is a set of consecutive even integers. What is the 29 Mar 2011, 00:39
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To add on top of what gmat1220 has mentioned, assume that you have a set of 39 consecutive even numbers, then the median = mean and thus two numbers on either side of mean will be 2 away, the next two will be 4 away, and so forth. So this gives the number of distances from mean, and the number of terms is already there, so you can substitute and find the SD.( Of course this is a SD question, so no need to calculate any further).


For option 2, there is nothing that gives how the other members of set are placed with respect to mean, or how many numbers are there in the set, and hence there is no definitive answer.

e.g. there could be three members (whereby the SD will be very low), or 100 members, in which case the SD will vary widely as many elements away from the mean will cause the SD to increase.
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There is a set of consecutive even integers. What is the 29 Mar 2011, 04:05
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thanks a ton guys...ok so statement makes sense, i got that right but my confusion was over statement 2

stmt 2 : it gives us the mean. We already know the that set is set of consecutive even numbers so we know it is evenly distributed so why cant we calculate the SD.

Can someone clarify this.
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There is a set of consecutive even integers. What is the 29 Mar 2011, 04:19
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thanks a ton guys...ok so statement makes sense, i got that right but my confusion was over statement 2

stmt 2 : it gives us the mean. We already know the that set is set of consecutive even numbers so we know it is evenly distributed so why cant we calculate the SD.

Can someone clarify this.
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380,382,384. Mean=382. Standard deviation: 2
378,380,382,384,386. Mean=382: Standard deviation: Somewhere between 2 and 4
376,378,380,382,384,386,388; Mean=382: Standard deviation: Somewhere between 4 and 6

Even though the even numbers are symmetrically distributed about the mean, with every inclusion of a pair of numbers, the standard deviation will gradually increase.

380,382,384:
Here there are just two numbers on both sides of 382. The deviation of 380 from the mean 382 is 2; the deviation of 384 from the mean 382 is also 2, thus the standard deviation is 2.

378,380,382,384,386
Here there are 5 numbers.
382 is the mean.
The deviation of 380 from the mean 382 is 2; the deviation of 384 from the mean 382 is also 2;
But,
The deviation of 378 from the mean 382 is 4; the deviation of 384 from the mean 386 is also 4;
Thus the standard deviation will be somewhere between 2 and 4.

Also, as per the rule, the standard deviation increases with increase in the Range of the set.
380,382,384. Range=4
378,380,382,384,386. Range=8
376,378,380,382,384,386,388. Range=12
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There is a set of consecutive even integers. What is the 29 Mar 2011, 04:37
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so stmt 2 is based on the concept of the more numbers you add closer to the mean ....sd changes.

thanks fluke.
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380,382,384. Mean=382. Standard deviation: 2
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Just to clarify, the standard deviation of that set is not equal to 2. If everything in a set is 2 away from the mean, the standard deviation will indeed be 2, but in your example, 382 is not 2 away from the mean; it is equal to the mean. Because of that, the standard deviation will certainly be less than 2.

If you care to complete the calculation, the distances of the elements in the set {380, 382, 384} to the mean are 2, 0 and 2. Squaring these and averaging, we get 8/3, so the standard deviation is sqrt(8/3).
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There is a set of consecutive even integers. What is the 29 Mar 2011, 10:14
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fluke

380,382,384. Mean=382. Standard deviation: 2

Just to clarify, the standard deviation of that set is not equal to 2. If everything in a set is 2 away from the mean, the standard deviation will indeed be 2, but in your example, 382 is not 2 away from the mean; it is equal to the mean. Because of that, the standard deviation will certainly be less than 2.

If you care to complete the calculation, the distances of the elements in the set {380, 382, 384} to the mean are 2, 0 and 2. Squaring these and averaging, we get 8/3, so the standard deviation is sqrt(8/3).
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True. My mistake. Thanks.
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