Set A = {1, 2, 3, 4, 5, 6, y} Which of the following possible values

A little correction/clarification

- A new value included in set which is closer to the mean than the value of standard deviation REDUCES standard deviation

- A new value included in set which is at an equal distance from the mean as the value of standard deviation is causes NO CHANGE standard deviation


- A new value included in set which is at a greater distance from the mean than the value of standard deviation INCREASES the standard deviation

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Closer to the mean and lesser the deviation. If equal to the mean then zero deviation.

for the set 1,2,3,4,5,6 ---> mean is 3.5

If we add y as 3.5 then there will be no deviation. hence D

Some background about Standard Deviation on the GMAT

For the purposes of the GMAT, it's sufficient to think of Standard Deviation as the Average Distance from the Mean. Here's what I mean:

Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
The mean of set A = 10 and the mean of set B = 10
How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

Alternatively, let's examine the Average Distance from the Mean for each set.

Set A {7,9,10,14}
Mean = 10
7 is a distance of 3 from the mean of 10
9 is a distance of 1 from the mean of 10
10 is a distance of 0 from the mean of 10
14 is a distance of 4 from the mean of 10
So, the average distance from the mean = (3+1+0+4)/4 = 2

B {1,8,13,18}
Mean = 10
1 is a distance of 9 from the mean of 10
8 is a distance of 2 from the mean of 10
13 is a distance of 3 from the mean of 10
18 is a distance of 8 from the mean of 10
So, the average distance from the mean = (9+2+3+8)/4 = 5.5

IMPORTANT: I'm not saying that the Standard Deviation of set A equals 2, and I'm not saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A.
More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GMAT.

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In the given question, the set {1, 2, 3, 4, 5, 6} has a mean of 3.5

Our goal is to determine the value of y that, when added to the set" will result in the smallest standard deviation.

If y = 3.5, this new value is a distance of 0 from the mean.
So, this y-value will result in the new set having the smallest standard deviation.

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Let’s first calculate the mean of the given 6 numbers:

(1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 7/2 = 3.5

We may recall that the closer the values in our set are to the mean, the smaller the standard deviation. Thus, when selecting a value for y, we want to select the value that will be closest to 3.5.

Thus, we see that choice D, 3.5, is correct choice.

Answer: D

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