The 10 students in a history class recently took an examination. What

The standard deviation is not the maximum deviation. The standard deviation is, while not exactly, basically the average deviation.

Since different sets of numbers could have the same standard (average) deviation from the mean, the mean and the standard deviation are not information sufficient for calculating the maximum score.

For example:

If the scores were 70, 70, 70, 70, 70, 80, 80, 80, 80, 80. The mean would be 75 and the standard deviation would be 5.

In this case the maximum is 80.

Meanwhile, if the scores were 69, 69, 69, 73, 74, 76, 77, 81, 81, 81, the mean would again be 75 and the standard deviation would be approximately 5.

So, we have the same mean and standard deviation, but the maximum is now 81.
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Hello Mo2men

I will try to explain. If 75 is the mean and SD is 5, it means that on average, every value deviates from mean by 5 units. That doesnt mean every value necessarily deviates from mean by 5 units. Its possible that few values deviate from mean by 8 or 10 units also but other values deviate from mean by 1 or 2 units only. Lets take example.

{70, 70, 80, 80}. For this set, mean is 75, and since every value deviates from mean by 5 only, SD of this set is also 5. And here the MAX value is 80.

Now consider {65, 75, 75, 75, 75, 75, 75, 85}. For this set, mean is 75, SD is 5 only. But here the MAX value is 85.

As the question stem doesn't make it clear where to start, we'll first try a few examples.
This is an Alternative approach.

(1) OK, but how much is the maximum? Could be 80, 85, 90, anything above the mean.
Insufficient.

(2) Still no useful information.... though finding specific examples is time consuming (as the equation for std is involved), we can think of the std as an expression with 10 variables representing the 10 different scores. So the maximum could, as above, be any number, such as 50, 30, 90, etc...
Insufficient.

Combined: Still not helpful, as we now have two equations for 10 variables (our 10 scores): std = 5 and mean = 75. So we cannot solve.

(E) is our answer.
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HI EgmatQuantExpert , GMATGuruNY

Can you please help me with this problem?

Standard deviation of the scores was 5 means it is 5 score variation, So if we combine 1 & 2 Mean = 75 Highest might be 80 & Least might be 70?
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For the purposes of the GMAT, we can consider standard deviation to be the AVERAGE DISTANCE FROM THE MEAN.



Statement 1 implies that the sum of the 10 scores = 10*75 = 750.
Statement 2 implies that the average distance from the mean = 5.
Given these two facts, we cannot determine the maximum score.
INSUFFICIENT.


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GMATGuruNY

1- Why the distance (mean) is not enough? Can't we say max = 80?

2-Can you please shed light when the two statement would be sufficient? what is the missing info here?

Thanks

Solution:

Given: 10 students in a history class recently took an examination
To find The maximum score on the examination.

Analysis of statement 1: The mean of the scores was 75.
This statement gives information about the mean value, but the maximum value can be anything greater than the mean value. i.e. it can be 85; 95; 105 etc…
Hence statement 1 is not sufficient to answer. We can eliminate options A and D.

Analysis of statement 2: The standard deviation of the scores was 5.
This statement gives information about on average every value deviates from mean by 5 units. This doesn’t mean that all the 10 values will be deviating from mean by 5 units; it’s also possible some may deviate 7 units or 8 units and some may deviate from mean by 2 or 3 units only.
Hence statement 2 is not sufficient to answer. We can eliminate option B.

Combining the statements 1 and 2; we get:
From statement 1; Mean value was 75
From statement 2; Standard deviation of the score was 5.
Combining two statements too we cannot answer the question.

The correct answer option is “E”.

The 10 students in a history class recently took an examination. What was the maximum score on the examination?

(1) The mean of the scores was 75.
(2) The standard deviation of the scores was 5.

Guys could you explain to me why E is the right answer here? I understood the SD as the Deviation from the mean. And wouldn't 75+5 be the max deviation possible?

Posted from my mobile device

Mean tells you the average and SD tells you how far or close to the mean the distribution of values is. Neither tells you what the ACTUAL maximum or minimum value is. You can have mean 75 and SD 5 in various different distributions.

70, 70, 70, 70, 70, 80, 80, 80, 80, 80
68,69,69,73,74,76, 77, 81,81,82
67,69,70,74,75,75, 76, 80,81,83

We don't know which is these is the actual case and hence we cannot say what the actual greatest score is. Hence answer is (E).


There is a maximum value that the greatest score can take under these constraints. That will happen in case of this list:

(75 - a), 75, 75, 75, 75, 75, 75, 75, 75, (75 + a)

Most values are at mean so that difference from mean is minimised. Most of the deviation will come from the maximum value but since we need the mean to be 75, we need to balance out the maximum value with the minimum value.

\(Sd = 5 = \sqrt{2a^2/10}\)
a = 11.18

So the maximum value that any one number can take is 75 + 11.18 = 86.18

In this case too
63.82, 75,75,75,75,75,75,75,75,86.18
mean = 75, SD = 5

But here is the thing - the question does not ask you the maximum value any number CAN take. The question only asks you the maximum value it does take. We cannot say what that is because the actual scenario could be any one of the 4 given above or any one of the many many others possible (within these constraints).

Answer (E)
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We have total value and standard deviation but we don't know each number and its distribution pattern.

The answer is C.
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Video solution from Quant Reasoning:
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Answer: Option E

Video solution by GMATinsight



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the reply from MartyTargetTestPrep is just more than awesome! thanks a lot ..
do we need to memorize the standard deviation formula for any gmat related need that can pop up in the exam? or it's enough to understand the actual meaning of this term?

You won't need the standard deviation formula. You need just to understand standard deviation conceptually.
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