For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
A 74
B 76
C 78
D 80
E 82
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Mean, Standard deviation
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Anurag@Gurome
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Say, mean score = M and standard deviation = Dpsm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
Hence, 58 = M - 2S and 98 = M + 3S
So, (M + 3S) - (M - 2S) = 98 - 58
--> 5S = 40
--> S = 8
Hence, M = (58 + 2S) = (58 + 2*8) = (58 + 16) = 74
The correct answer is A.
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Brent@GMATPrepNow
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A little background on units of standard deviation:psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
A 74
B 76
C 78
D 80
E 82
If the SD is 4, then 1 unit of SD = 4
Similarly, 2 units of SD = 8
And 1.5 units of SD = 6
If the mean is 10 and the SD is 4, then we say that 18 is 2 units of SD above the mean since 10 + 2(4) = 18
Similarly, we say that 6 is 1 unit of SD below the mean since 10 - 4 = 6
For your question, we can let M=mean and let D=the standard deviation
So, 58 is 2 standard deviations below the mean translates into M - 2D = 58
and 98 is 3 standard deviations above the mean translates into M + 3D = 98
When we solve this system of equations, we get M=74 and D=8
So the answer is A
Cheers,
Brent
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Ian Stewart
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There's no need to solve a two equations/two unknowns problem here; if 58 and 98 are five standard deviations apart, the standard deviation is (98-58)/5 = 8. Thus the mean is 58 + (2)(8) = 74.psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
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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The Iceman
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Another perspective.psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
A 74
B 76
C 78
D 80
E 82
Imagine this sentence appears as the last question on the GMAT and you have only 15-20 secs to mark the answer.
Just take the avg of the two numbers avg(58,98) = 78. Now our ans must be less than 78 as the mean has to be farther from 98 and closer to 58.
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shreerajp99
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If we start with ans choices here,say we consider C(78), we find that 98 and 58 are 20 apart from 78.This cant be our mean.
We go to 74 and see that ok,2 std deviations below mean=58 and 3 above mean = 98.So the mean is 74 and std deviation is 8.
Thanks,
Shreeraj
We go to 74 and see that ok,2 std deviations below mean=58 and 3 above mean = 98.So the mean is 74 and std deviation is 8.
Thanks,
Shreeraj
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cking6178
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Iceman, your method still leaves 2 answer choices...Ian solved it exactly the way I did & it took me less than a minute to work through the problem (ie...allows you to solve the problem efficiently, giving you more time to work on complicated problems)...Assuming that you understand what standard deviation means, you can simply obtain the range, which is 40 (98-58 = 40) and we know that we are 2 sd below and 3 above, so that means we have a 5 sd spread which we divide into 40, giving us a sd of 8. Verify this works by subtracting sd*3 (24) from the high score of 98 and making sure that is equal to adding sd*2 (16) to the low score of 58 and you get 74 in both cases, so the answer is A.The Iceman wrote:Another perspective.psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
A 74
B 76
C 78
D 80
E 82
Imagine this sentence appears as the last question on the GMAT and you have only 15-20 secs to mark the answer.
Just take the avg of the two numbers avg(58,98) = 78. Now our ans must be less than 78 as the mean has to be farther from 98 and closer to 58.
The explanation is wordy, but the math is quick and painless.
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The Iceman
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Cking6178, I think you misinterpreted me. I never provided the solution anyway. All i told was that if faced in a time bound situation you could easily get rid of 3 options and make an educated guess.cking6178 wrote:Iceman, your method still leaves 2 answer choices...Ian solved it exactly the way I did & it took me less than a minute to work through the problem (ie...allows you to solve the problem efficiently, giving you more time to work on complicated problems)...Assuming that you understand what standard deviation means, you can simply obtain the range, which is 40 (98-58 = 40) and we know that we are 2 sd below and 3 above, so that means we have a 5 sd spread which we divide into 40, giving us a sd of 8. Verify this works by subtracting sd*3 (24) from the high score of 98 and making sure that is equal to adding sd*2 (16) to the low score of 58 and you get 74 in both cases, so the answer is A.The Iceman wrote:Another perspective.psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
A 74
B 76
C 78
D 80
E 82
Imagine this sentence appears as the last question on the GMAT and you have only 15-20 secs to mark the answer.
Just take the avg of the two numbers avg(58,98) = 78. Now our ans must be less than 78 as the mean has to be farther from 98 and closer to 58.
The explanation is wordy, but the math is quick and painless.
By the way my solution is the same as Ian's
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Brent@GMATPrepNow
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Great solution, Ian. I was totally fixated on the algebraic approach. Fortunately, that approach isn't much longer than just applying some common sense (as you did).Ian Stewart wrote:There's no need to solve a two equations/two unknowns problem here; if 58 and 98 are five standard deviations apart, the standard deviation is (98-58)/5 = 8. Thus the mean is 58 + (2)(8) = 74.psm12se wrote:For a certain examination, a score of 58 was 2 standard deviation below the mean and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination?
Welcome back!
Cheers,
Brent
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lunarpower
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notice one important theme at work here:
if you're given a numerical value of the standard deviation, it doesn't even matter that it's called "the standard deviation".
let me illustrate.
in this problem, let's remove all references to the standard deviation, and instead refer to it as the "pink flamingo".
then we have:
98 is 3 pink flamingoes above the mean --> 98 = mean + 3(PF)
58 is 2 pink flamingoes below the mean --> 58 = mean - 2(PF)
subtract these two equations:
98 - 58 = 5(PF)
40 = 5(PF)
8 = pink flamingo
the rest follows.
this is an interesting twist on the standard deviation: there are LOTS of gmatprep problems on which the value of the standard deviation is specified, and, uncannily enough, not one of those problems requires an actual understanding of what "standard deviation" means.
instead, on ALL of them, all you have to do is treat the SD as though it were some other random quantity (like "pink flamingo").
by contrast, on problems featuring a standard deviation that's NOT given a numerical value - such as problems on which you have to figure out whether the addition of certain numbers to a set will increase or decrease the standard deviation, without actually knowing the value of the standard deviation itself - you actually do need to understand the conceptual significance of the standard deviation itself.
if you're given a numerical value of the standard deviation, it doesn't even matter that it's called "the standard deviation".
let me illustrate.
in this problem, let's remove all references to the standard deviation, and instead refer to it as the "pink flamingo".
then we have:
98 is 3 pink flamingoes above the mean --> 98 = mean + 3(PF)
58 is 2 pink flamingoes below the mean --> 58 = mean - 2(PF)
subtract these two equations:
98 - 58 = 5(PF)
40 = 5(PF)
8 = pink flamingo
the rest follows.
this is an interesting twist on the standard deviation: there are LOTS of gmatprep problems on which the value of the standard deviation is specified, and, uncannily enough, not one of those problems requires an actual understanding of what "standard deviation" means.
instead, on ALL of them, all you have to do is treat the SD as though it were some other random quantity (like "pink flamingo").
by contrast, on problems featuring a standard deviation that's NOT given a numerical value - such as problems on which you have to figure out whether the addition of certain numbers to a set will increase or decrease the standard deviation, without actually knowing the value of the standard deviation itself - you actually do need to understand the conceptual significance of the standard deviation itself.
Ron has been teaching various standardized tests for 20 years.
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On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi
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buzzdeepak
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Here is how I solved (I like to "see" things and try to draw pictures or tables as and when I can - nothing fancy, just quick and dirty)
[/img]
So, 5x = 98-58 = 40
x = 8
Therefore, 2x = 16
The answer, therefore, is 58+16 = 74
Nothing different than what all the experts have said earlier, just another way (sometimes a picture says a thousand words)
[/img]
So, 5x = 98-58 = 40
x = 8
Therefore, 2x = 16
The answer, therefore, is 58+16 = 74
Nothing different than what all the experts have said earlier, just another way (sometimes a picture says a thousand words)
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ngalinh
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Yes, I like your image, especially if you turn it so that 2x is below 3xbuzzdeepak wrote:Here is how I solved (I like to "see" things and try to draw pictures or tables as and when I can - nothing fancy, just quick and dirty)
[/img]
So, 5x = 98-58 = 40
x = 8
Therefore, 2x = 16
The answer, therefore, is 58+16 = 74
Nothing different than what all the experts have said earlier, just another way (sometimes a picture says a thousand words)
