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Gmat Prep - Standard Deviation and mean

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by dubeystuts » Sat Jan 31, 2009 4:24 am
70, 75, 80, 85, 90, 105, 105, 130, 130, 130
List shown consist of the times, in seconds, that it took each of 10 school children to run a distance of 400 meters. If the standard deviation of the 10 running times is 22.4 seconds, rounded to the nearest tenth of the second, how many of the 10 running times are more than 1 standard deviation below the mean of the 10 running times?

A) 1
B) 2
C) 3
D) 4
E) 5

Can someone please explain what is the best aproach to answer this question? Correct answer is B.
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by DanaJ » Sat Jan 31, 2009 4:34 am
First of all, you should calculate the mean of these numbers. That should be pretty easy to do, since, as you can see, the numbers are 5, 10, 15 or 25 apart. So let's start by noticing that each number equals 70 + n*5. Let me give you some examples:
70 = 70 + 0*5
75 = 70 + 1*5
130 = 70 + 12*5
So you get that the sum of the numbers is 10 times 70 (since you have 10 kids) plus 5(0 + 1 + 2 + 3 + 4 + 7 + 7 + 12 + 12 + 12) = 5*60. This means that the mean will be (10*70 + 5*60)/10 = 70 + 5*6 = 100.
Now, having a std dev of 22.4 means that the numbers in your series are an average of 22.4 apart from the mean. You are looking for running times that are more than 1 std dev smaller than the mean, which is to say 100 - 22.4 = 77.6. There are only two running times smaller than 77.6, and those are 70 and 75. So the answer is B.
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