For a certain exam, a score of 58 was 2 standard deviations below mean

Question

For a certain exam, a score of 58 was 2 standard deviations below mean and a score of 98 was 3 standard deviations above mean. What was the mean score for the exam?

A. 74
B. 76
C. 78
D. 80
E. 82

Bunuel · Accepted Answer

KarishmaB · Answer

These are concepts of normal distribution. You don't need to delve much into details but you must understand what 1 SD, 2 SD etc mean.

Say you have a distribution of weights in a class of children. The mean weight of the class is, say 46 kgs. If I tell you that 1 standard deviation is of 5 kgs, it means that 1 SD below mean is 5 kgs less than 46 kgs which is 41 kgs. This also means that 34% children have weight ranging from 46 kgs to 41 kgs but you don't have to worry about that.

So 2 SD below mean will be 2*5 = 10 kgs less than 46 i.e. 36 kgs.

Similarly, 3 SD above mean will be 3*5 = 15 kgs more than 46 i.e. 61 kgs.

In the original question, you are given that 58 is 2 SD below mean and 98 is 3 SD above mean. So from 58 to 98, we have 5 SDs. 
 \frac{{98 - 58}}{5} = 8  = SD

Since SD is 8 so mean is at  58 + 2*8 = 74

aabougabal · Answer

Quick solution --> please see below
58=M-2D
98=M+3D
98-58=M+3D-M-2D
40=5D
D=8 so, One SD =8

58=M-2D
58+2D=M
58+(2x8)=M
58+16=M
So M = 74

apply the same for the other equation and you will get to M=74.

Thanks!

patternpandora · Answer

Welcome to GMAT Club. Below is a solution to your question.

A score of 58 was 2 standard deviations below the mean --> 58 = Mean - 2d
A score of 98 was 3 standard deviations above the mean --> 98 = Mean + 3d

Solving above for Mean = 74.

Answer: A.

Can someone please elaborate as how is this solved?

gmat6nplus1 · Answer

you can either solve as Bunuel did or exploit number properties and plug in from the answer choices:

58 = Mean - 2d -----> 2d = Mean - 58 
98 = Mean + 3d -----> 3d = 98 - Mean

Thus your d must be both a multiple of 2 (2n) or a multiple of 3 (2n+1). Giving a fast glance at the answer choices, since E-E = E, the first equation is satisfied in every answer choice. The second equation, however, is satisfied in two answers only, respectively A and D.

Now, since d is a unique value for both equations, calculating d must yield the same result. It does but for one choice only and that choice is A.

d=\frac{74-58}{2}=\frac{98-74}{3}

This is intended to be a back-up method, for sure not as fast as that one Bunuel came up with but when you're under pressure everything that pops up in your mind is useful.

kanusha · Answer

Every list of numbers has a mean. every number on the list has a deviation from the mean: that is how far that number is from the mean.
deviation from the mean = (value) – (mean)
Or
Value= Mean - Deviation
58 = M - 2d
98 = M - 3d
Solve it
M = 74
:-D

JackSparr0w · Answer

For those confused like myself, 58=M-2d and 98=M+3d is a system of equations. We want the value of M, so isolate D in one of the equations. then plug that value of d into the other equation and solve for M.

manishbhusal · Answer

I am getting m=-22. I will be glad is someone point out the mistake i have committed.

58=m-2d 1
98=m-3d 2

subtracting 2 from 1

d=-40.

substituting d in 1

m =-22.

chetan2u · Answer

hi  manishbhusal ,
98 is 3 deviation above ..
so 98=m+3d and not m-3d..

LogicGuru1 · Answer

Mean -2 SD = 58 ..... eq 1 
Mean +3SD =98 ........eq 2

Subtract eq 2 from eq1 
5SD= 40
SD = 40/5--> 8

So the standard deviation is 8 
now to find the mean 
either add 2*SD to 58 = 2*8+58 = 18+58=74
Or subtract 3*SD from 98 = 98-3*8 = 98-24 = 74
THE ANSWER IS DEFINITELY A = 74

SaeidH · Answer

The easiest approach ever: 
98-58=40
there are 5 SD in between; hence, 
40/5=8
58+2(SD) i.e. 58+16=74
or 
98-3(SD) i.e. 98-24=74

akadiyan · Answer

For a certain exam,a score of 58 was 2 standard deviations below mean and a score of 98 was 3 standard deviations above mean.What was the mean score for the exam?

The range in given question is 98-58 = 40

58 is 2 SD below mean
90 is 3 SD above mean

So there is 5 SD in between 58 and 98. 5 SD=40 , SD = 8

The mean score is 74
Ans :  A

BrentGMATPrepNow · Answer

----------ASIDE---------------------
A little extra background on   standard deviations above and below the mean

If, for example, a set has a standard deviation of 4, then: 
1 standard deviation = 4 
2 standard deviations = 8 
3 standard deviations = 12 
1.5 standard deviations = 6 
0.25 standard deviations = 1 
etc

So, if the mean of a set is 9, and the standard deviation is 4, then: 
2 standard deviations ABOVE the mean =  17    17 ]  
1.5 standard deviations BELOW the mean =  3    3 ]  
3 standard deviations ABOVE the mean =  21    21 ]  
etc.
-------------------------------------------------------------

For this 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

JeffTargetTestPrep · Answer

If we let x = mean and s = standard deviation, then a score of 58 that was 2 standard deviations below the mean would mean 58 = x – 2s. Similarly, a score of 98 that was 3 standard deviations above the mean would mean 98 = x + 3s.
 
We can use the two equations to determine a value of x.
 
x – 2s = 58
 
x + 3s = 98
 
If we subtract the first equation from the second equation, we have:
 
5s = 40
 
s = 8
 
Now we can determine x:
 
x + 3s = 98
 
x + 3(8) = 98
 
x + 24 = 98
 
x = 74
 
Answer: A

minustark · Answer

m -2d =58
m+3d = 98
5d = 40
d =8
So, mean = 98 - 3*8 = 74

MHIKER · Answer

m+3d=98---I

m-2d=58---II

Substracting II from I: m+3d-m+2d=98-58

=5d=40

d=5

Substituting the value in II
m-16=58

m=74

The answer is A.

Telo · Answer

Thank you, I messed up while solving for these equations.(I know, such a silly mistake, but a costly one!) Your quick solution helped me so much, thanks.

Thanks!

bamfromtowerofgod · Answer

Hello, my understanding is that in system of equations, when applying elimination, we need to add the two equations. Why are you subtracting the first equation from the second?

chetan2u · Answer

When you are dealing with equations ( where sign ‘=‘ is used), you can add/subtract/multiply. 
But when you are dealing with inequalities ( where sign ‘<,>’ are used), you can only add the same sides of inequality.

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