Standard deviation question

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okletsdothis: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Fri Aug 13, 2010 8:13 am; GMAT Score:650

by okletsdothis » Thu Jan 27, 2011 6:47 am

I got confused in the wordings.

Some body please explain

The standard deviation of a diver's score is calculated by finding the average (arithmetic mean) of the squared differences of each score from the average (arithmetic mean) of all the scores. An Olympic diver received the following scores: 6.0, 5.5, 7.0, 6.5, and 5.0. The standard deviation of her scores is in which of the following ranges?

A. 0 - 1.9
B. 2 - 3.9
C. 4 - 6.9
D. 7 - 7.9
E. 8 - 9.9

OA A

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Source: Beat The GMAT — Problem Solving | Thu Jan 27, 2011 6:47 am

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by Anurag@Gurome » Thu Jan 27, 2011 7:18 am

okletsdothis wrote:The standard deviation of a diver's score is calculated by finding the average (arithmetic mean) of the squared differences of each score from the average (arithmetic mean) of all the scores. An Olympic diver received the following scores: 6.0, 5.5, 7.0, 6.5, and 5.0. The standard deviation of her scores is in which of the following ranges?

A. 0 - 1.9
B. 2 - 3.9
C. 4 - 6.9
D. 7 - 7.9
E. 8 - 9.9

Average of the scores = 6.0

Difference of the scores from the average are : 0, 0.5, 1.0, 0.5, and 1.0
Square of the differences are : 0, 0.25, 1.0, 0.25, and 1.0

Sum of the square of the differences = (0.25 + 0.25 + 1.0 + 1.0) = 2.5

Average of the square of the differences = 2.5/5 = 0.5

The correct answer is A.

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aleph777: Master | Next Rank: 500 Posts; Posts: 131; Joined: Fri Jun 18, 2010 10:19 am; Location: New York, NY; Thanked: 10 times

by aleph777 » Thu Jan 27, 2011 7:32 am

If you're not too familiar with Standard Deviation, the question was kind enough to explain how to actually find it, although it's written in convoluted terms. I find the best way to approach a complex word problem like that is to break it down slowly into segments.

The question asks for the standard deviation (SD) of the diver's score, and says you can find the SD by finding the average of the squared differences of each score from the average of all scores.

Let's work backwards and look at that question in smaller bits.

In order to find anything at all, we need the average of all scores. And we know that average = sum of all scores / total number of scores. In this case, since there are five scores, we know the average is just the middle score, which is 6.0.

So that's our starting point.

Next, we need to find the difference of each score from the average. Now that we know the average is 6.0, we need to find the difference between the average and each score. So that's simple subtraction:

6.0 - 5.0 = 1.0
6.0 - 5.5 = 0.5
6.0 - 6.0 = 0.0
6.0 - 6.5 = -0.5
6.0 - 7.0 = -1.0

Next, it says we need to find the average of the squared differences. We just got the difference between each number and the average, so now we need to make a new average of each difference squared. So we return to the average formula and square each term.

A = (1^2 + .5^2 + 0^2 + (-.5)^2 + (-1)^2)/5 = (1 + .25 + 0 + .25 + 1)/5 = 2.5/5 = .5

And that's the SD.

Therefore, the answer is A.

Hope that helps!

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okletsdothis: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Fri Aug 13, 2010 8:13 am; GMAT Score:650

by okletsdothis » Thu Jan 27, 2011 9:55 am

thnk u both. tht was really helpfull.

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prachich1987: Legendary Member; Posts: 752; Joined: Sun Sep 12, 2010 2:47 amFollowed by: 10 members; Thanked: 20 times; GMAT Score:700

by prachich1987 » Fri Jan 28, 2011 2:39 am

Anurag@Gurome wrote:
okletsdothis wrote:The standard deviation of a diver's score is calculated by finding the average (arithmetic mean) of the squared differences of each score from the average (arithmetic mean) of all the scores. An Olympic diver received the following scores: 6.0, 5.5, 7.0, 6.5, and 5.0. The standard deviation of her scores is in which of the following ranges?

A. 0 - 1.9
B. 2 - 3.9
C. 4 - 6.9
D. 7 - 7.9
E. 8 - 9.9
Average of the scores = 6.0

Difference of the scores from the average are : 0, 0.5, 1.0, 0.5, and 1.0
Square of the differences are : 0, 0.25, 1.0, 0.25, and 1.0

Sum of the square of the differences = (0.25 + 0.25 + 1.0 + 1.0) = 2.5

Average of the square of the differences = 2.5/5 = 0.5

The correct answer is A.

Hi Anurag,

Just wanted to confirm what you have calculated is variance
For calculating the standard deviation we need find square root of 0.5
Am I right?

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Fri Jan 28, 2011 3:01 am

prachich1987 wrote:Hi Anurag,

Just wanted to confirm what you have calculated is variance
For calculating the standard deviation we need find square root of 0.5
Am I right?

Yes, if you go with standard definition of standard deviation then you should take the square root of 0.5. But please note that this problem has defined standard deviation in its own words. According to the problem the average of the squared differences of each score from the average is the standard deviation. That's why 0.5 is the standard deviation as far as this particular problem is concerned.

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prachich1987: Legendary Member; Posts: 752; Joined: Sun Sep 12, 2010 2:47 amFollowed by: 10 members; Thanked: 20 times; GMAT Score:700

by prachich1987 » Fri Jan 28, 2011 3:04 am

Anurag@Gurome wrote:
prachich1987 wrote:Hi Anurag,

Just wanted to confirm what you have calculated is variance
For calculating the standard deviation we need find square root of 0.5
Am I right?
Yes, if you go with standard definition of standard deviation then you should take the square root of 0.5. But please note that this problem has defined standard deviation in its own words. According to the problem the average of the squared differences of each score from the average is the standard deviation. That's why 0.5 is the standard deviation as far as this particular problem is concerned.

Thanks Anurag
So for this problem they have given the definition for S.D and hence we will follow that
Otherwise if they had not given any such definition we would have followed the actual definition
that is we would have taken the square root of 0.5

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Fri Jan 28, 2011 3:13 am

prachich1987 wrote:So for this problem they have given the definition for S.D and hence we will follow that
Otherwise if they had not given any such definition we would have followed the actual definition
that is we would have taken the square root of 0.5

Yes.

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