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Standard deviation and mean

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by Ian Stewart » Sun Nov 09, 2008 4:16 am
StarDust845 wrote: Here we go. This problem is from "The Princeton Review. Cracking the GMAT, Bin4 hard problems).

The members of the newest recruiting class of a certain military organization are taking their physical conditioning test, and those who score in the bottom 16 percent will have to retest. If the scores are normally distributed and have an arithmetic mean of 72, what is the score at or below which the recruits will have to retest?

(1) There are 500 recruits in the class.
(2) 10 recruits scored 82 or higher.
This is not a well-designed question. The normal distribution is a continuous distribution; that is, it describes infinite sets. A finite set of data can only be 'approximately normal'. It also seems that their solution is incorrect. Clearly they intend for the standard deviation to be 5, but in normally distributed data, roughly 2.3 percent will be two or more standard deviations above the mean, so for the standard deviation to be 5, Statement 2 should read "11 recruits scored 82 or higher" (actually, it should be closer to 11.5).

So the question is poorly designed for a start, and the solution provided isn't quite correct. I've never seen a real GMAT question that requires you to know anything about normally distributed data either (if anyone has, please let me know!), so the question above is not at all important to understand for the test.
Stuart Kovinsky wrote:I haven't seen every GMAT question on every test in GMAT history, but I've neither seen nor heard of one that required you to calculate standard deviation (or to know the exact ranges of 1SD, 2 SD, etc...).
We can only know the exact percentages lying within, say, two standard deviations of the mean if we know how the data is distributed, of course; it is not typically true that 95.4% of data is within two standard deviations of the mean unless the data is normally distributed. So knowing 'the exact ranges' would only be helpful on a question specifically about normally distributed data, and to my knowledge, those never appear on the GMAT.
Stuart Kovinsky wrote:The only standard deviation questions of which I know require you to understand what SD is (these questions usually appear in DS) and to be able to tell the difference between a set with low SD and one with high SD (can appear in PS or DS).
Yes, exactly, along with questions that require you to know how standard deviation is discussed -- questions like "If the mean of a set of data is 700 and the standard deviation is 120, what value is more than two standard deviations away from the mean?"
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by MTeles » Wed Jul 22, 2009 6:10 pm
?
I did not understand.
Any one can be more didatic?
:cry:
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by tarn151 » Sat Aug 07, 2010 11:12 am
MTeles wrote:?
I did not understand.
Any one can be more didatic?
:cry:
MTeles, though I agree with the pro's on here that this is a bad question and lacks a few points, here is how to answer this one.

So I figured this one out... it's definitely a tough one and I wouldn't recommend trying to spend time to solve it out on the actual exam. However, knowing that these pieces of information will find you the answer could help on the exam.

Anyhow, here's how to answer this one...

I'm going to skip past A) and B), because it's clear that they don't provide enough info. So by combining them, we get the following:

10 of the 500 recruits scored 82 or higher. 10/500= 2%

Now, lets take a look at a Normal Distribution chart:

ImageImage

This graph gives you a good idea of what % is covered by each "block" of standard deviation (1 up and 1 down). The general rule on this for the GMAT is that 1 standard deviation covers 68%, 2 cover 95%, 3 cover 99%. As you can see in the chart above, standard deviations cover the variance above and below. The chart above has slightly different percentages, but helps for you to understand the logic I'm about to make.

Point A) So, 1 standard deviation above equals ~34%, 2 above equals ~13.5%.. (95%-68%)/2, and 3 above covers ~2%... (99%-95%)/2. The rest, is covered by .5%

Point B) Since the mean is 72 and 3 standard deviations from the mean begins at 82, there are two standard deviations between 72 and 82. The difference here is 10, so each standard deviation represents a range of 5.

Point C) That means that 1 standard deviation covers 72-77, 2 standard deviations covers 77-82... and the opposite logic can be applied to the lower end of the data (-1 sd = 67-72, -2 sd = 62-67, and -3 = 57-62).

Point D) Now lets take a look at how the bottom 16% correlates to standard deviation. Per "point A", the bottom ranges cover ~13.5%, ~2%, and ~.5%. Add these three together and you get the bottom 16%. So, lets go back to "point c" and we get that this 16% falls into the score range of anything below 67 (anything below the 67-72 range of -1 sd). There's your answer!

I agree with others that this is a poorly stated question and makes you assume a few points (suchs as the approximate percentages that I gave for range of standard deviation), but this general logic will help if a similar question comes up on your exam. [/img]
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