Standard deviation table

AbeNeedsAnswers wrote:How do we calculate standard deviation?

---------------- Mon--Tues--Wed--Thu--
Company A---45-----55----50----50
Company B---10-----30----30----10
Company C---34-----28----28----30
Company D---39-----42----41----38
Company E---50-----60----60----70

The table above shows the number of packages shipped daily by each of the five companies during a 4-day period. The standard deviation of the numbers of packages shipped daily during the period was greatest for which of the five companies?

A) A
B) B
C) C
D) D
E) E

B

The standard deviation of a data set is a measure of the deviations of the numbers in the data set w.r.t. their mean. More deviated (more distant) the numbers are from the mean, the more is their standard deviation and vice-versa.

Let's calculate the mean of the five companies and the deviations w.r.t. their mean.

'Dev-M' means the deviation of Monday and similarly for others...

-------- Mon--Tues--Wed--Thu--Mean--Dev-M--Dev-T--Dev-W--Dev-Th
Co A---45-----55----50----50-----50-------5---------5---------0--------0
Co B---10-----30----30----10-----20------10--------10-------10------10
Co C---34-----28----28----30-----30-------4---------2---------2--------0
Co D---39-----42----41----38-----40-------1---------2---------1--------2
Co E---50-----60----60----70-----60-------10--------0--------0--------10

We see that each value (10 each) of deviations for Company B is greater than that for companies A, C, and D. Also the two values for Company E are 0, while the other two (10 each) are equal, thus, SD would be the greatest for Company B.

In the GMAT, there is no need to compute the SD, mere analysis suffices.

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Sets & Statistics Guide

-Jay
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Jay has done a great job solving the question.

I'd like to add that, 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.

Cheers,
Brent

Here are a few more practice questions where we can apply the concept of "average distance from the mean" as an approximation for Standard Deviation:

https://www.beatthegmat.com/standard-dev ... 74384.html
https://www.beatthegmat.com/standard-dev ... 69584.html
https://www.beatthegmat.com/range-and-sd-t89159.html