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Standard deviation and mean

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Standard deviation and mean

by StarDust845 » Fri Jan 11, 2008 11:48 am
How do we know that the top 2% of the scores lie more than 2 standard deviations apart from the mean in a normal distribution?

1) In one of the problems from hard bin 4 in Princeton review, the explanation of answer depended on the rule I stated above. Is it just an empirical result that I have to remember?

2) What is variance? Is it the square of standard deviation?

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by cris » Fri Jan 11, 2008 12:18 pm
Calista could you post the question?

Thanks!
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by StarDust845 » Fri Jan 11, 2008 1:36 pm
Moderator,

Is it okay to post the question and its source\author directly?

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Re: Standard deviation and mean

by rajmirra » Fri Jan 11, 2008 2:05 pm
StarDust845 wrote:How do we know that the top 2% of the scores lie more than 2 standard deviations apart from the mean in a normal distribution?

1) In one of the problems from hard bin 4 in Princeton review, the explanation of answer depended on the rule I stated above. Is it just an empirical result that I have to remember?

2) What is variance? Is it the square of standard deviation?

Calista.
Yes variance is square of Standard deviation.

Now lets look what variance is. Variance is the mean of the square of individual deviations from their mean. Confusing?

Okay, I have 10 entries with ceratin values ranging from 1 to 10. The mean of the 10 is 5. Entry 1 has a value 3. So it deviated -2 from the means and the square is +4. I have to do the rest of the 9 and find the average of the squares of this deviation.

The formula is sum of (X-x)Squared/n where Block X is Mean and lower case x is deviation and n is the total number of entries.

If you plot the 10 entries in an X axis the mean will be at the centre. A normal distribution curve will look like a inverted bell. It will increase at first reach a maximum and will drop down. So the highest score should be near the centre of the curve.

I am sorry I cannot bring a graph or sketch to emphasise here.

You can correct me if I am wrong.

Thanks
Raj
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by StarDust845 » Fri Jan 11, 2008 2:34 pm
rajmirra,

No, it isn't wrong. Your answer is correct for my question #2 and
I understand the variance now. Thanks for that.

I am waiting for moderator to respond to my question before I can post the official question.

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by Stuart@KaplanGMAT » Fri Jan 11, 2008 8:03 pm
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...).

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).
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by gabriel » Sat Jan 12, 2008 12:34 am
StarDust845 wrote:Moderator,

Is it okay to post the question and its source\author directly?

Calista.
Sorry for the late reply.. but that should be fine

Regards :)
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by StarDust845 » Sat Jan 12, 2008 7:25 am
(Moderator please move to DS area, I started by just asking a question but it is actually a DS problem).

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.

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by cris » Sun Jan 13, 2008 10:04 am
I think is E.

Because we know its some where between 62-72 but I dont think it can be found the exact score.
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by Stuart@KaplanGMAT » Sun Jan 13, 2008 10:30 am
I think (although I'm not sure, this is a much more detailed SD question than we usually see on the GMAT) that the answer is C (together).

Just the number of participants is irrelevant, since we don't know how big 1 SD is.

Just the number of people who scored 82 or above is irrelevant, because we don't know what % those people are of the entire group.

However, if we combine them, we know that 10/500 = 1/50 = 2% of recruits scored 82 or above. Since it's a normal distribution (info from the question stem), we know how many SDs from the mean this group of people should be, which means we can calculate the size of 1 SD. Once we know the size of 1 SD, we can calculate how far below the mean the bottom 16% of test takers should be.

Note: this would never appear as a problem solving questions, since we'd need too many details of SD to actually make the calculations.
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by cris » Sun Jan 13, 2008 11:56 am
So normal distribution measn that the same increasethat there is in the firts 2% occurs also in the next 2%??

I did not know that.
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by StarDust845 » Sun Jan 13, 2008 6:16 pm
cris wrote:So normal distribution measn that the same increasethat there is in the firts 2% occurs also in the next 2%??

I did not know that.
No. Refer to the "Standard deviation and confidence intervals" section in the following link https://en.wikipedia.org/wiki/Normal_distribution

It states: "About 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. This is known as the "68-95-99.7 rule" or the "empirical rule."

There you will see that the top 2% of the scores fall two standard deviations away from the mean. I think that is what the author of the question was referring to. But as Stuart said, may be this is too detailed a standard deviation question for GMAT.
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by yalephd2007 » Wed Apr 16, 2008 7:26 am
I will go with E too
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doing this question again!

by katty » Sun Oct 19, 2008 9:39 am
Princeton Review Question 3 in Hard Math Bin - this forum was discussing it previously and I am now doing the question and find it confusing. I get that the top 2% should be the 3rd standard deviation, but then the answer in the book says the scores below 67 and 62 (2 and 3 standard deviations from the mean) represent the bottom 16% of the class. I don't get that...
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by mental » Sat Nov 08, 2008 4:32 am
I will go with C

both together sufficient.........scores below 67 fall in bottom 16%
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