In a set of twenty numbers, 19 of the 20 numbers are between 40 and 50

Question

In a set of twenty numbers, 19 of the 20 numbers are between 40 and 50. Is the median greater than the mean?

(1) The standard deviation is greater than 15

(2) The 20th number is greater than 100

Kudos for a correct  solution .

Lucky2783 · Answer

(1) The standard deviation is greater than 15
 mean and median of the 19 elements are within 40-50.
 
 Standard deviation of 19 elements will be minimum (0) when all 19 elements are same .

Standard deviation of 19 elements will be maximum when there are 10 50s and 9 40s . Roughly 5.

adding 1 new element to this set has caused the standard deviation to go up .

2 cases ,

a) the 20th element is less then all the 19 elements. it can be -1000 , -2000 etc. Mean will move on left side with median standing taller than mean on the histogram.

b) the 20th element is greater then all the 19 elements. it can be 1000 or any big number. Mean will move on right side with mean greater than the median .

(2) The 20th number is greater than 100

as 100 is a large number compared to remaining 19 elements so the mean will move towards right by a big magnitude, causing median to be less than the mean.

Sufficient.

Answer should be B.

Harley1980 · Answer

1) we don't know about 20-th number. So it can be infinitely big and in this case mean will be greater than median or infinitely small and in this case mean will be smaller than median 
Insufficient

2) we know that 20-th number bigger than 100. Let's take such case:
first nine numbers = 41 next 10 numbers equal 49 and last number equal 101
Median = 49 and Mean = 48 so mean less than median
And if we take infinitely big last number than mean will be more than median
Insufficient

1+2) As we know that SD > 15 than 20-th number should be more than 101 and we know that mean will be bigger than median.
Sufficient
And answer is C

Bunuel · Answer

MAGOOSH OFFICIAL SOLUTION:

For a symmetrical distribution, the mean = median, and if the median is close to symmetry, the mean and the median are close in value. When the distribution of numbers is radical asymmetrical, with one outlier or several outliers on only one side of the distribution, then the mean is pulled in the direction of the outliers. The median,  resistant to outliers , stays in the middle of the majority of numbers, but the mean is  sensitive to outliers , gets pulled in their direction. High outliers pull the mean up, and low outliers pull the mean down.

Statement #1:  the standard deviation is greater than 15

This tells us that there’s large variation, suggesting that the 20th number is far away from the other 19, but far away in which direction? Much higher or much lower than the rest of the numbers? We don’t know. A high outlier would pull the mean up, and a low outlier would pull the mean down. Here, we know we have an outlier, but we don’t know its directions, so we don’t know in which direction the mean is affected. We cannot answer the question. This statement, alone and by itself, is not sufficient.

Statement #2:  the 20th number is greater than 100

Now, we know that the outlier is a high outlier, much bigger than the other numbers in the set. A high outlier pulls the mean up, away from the median, so the mean is higher than the median. We can give a definitive “yes” answer to the prompt question. This statement, alone and by itself, is sufficient.

Answer = (B)

godot53 · Answer

If we consider this following scenario
First 9 numbers: 41 X 9 = 369  
The second set of 10 numbers : 49 X 10 = 490

case- I
the last number is assumed to be 101 
so Mean = 48
Median = 49
Median > mean.
---
Case - II:
The last number is assumed to be 10,000
now, we can safely conclude that mean>median.
--
So, How B is the correct answer ? Where I am going wrong..

Edit 1 - grammatical error/spelling error

ganand · Answer

Hi  godot53 ,

I think your example is perfectly fine. I don't see any error in it. Let others comment on it.

Thank you.

emockus · Answer

Statement 1: No information is given on the 20th value making it impossible to determine the limit of the set, and thus the mean cannot be determined.

Statement 2: We are given that the limit of the set is greater than 100. This is sufficient as we can answer YES/NO on whether the median is greater than the mean.

godot53 · Answer

Hello  Bunuel  : It would be great if you can consider this.
 
I understand that there is some problem with this question.

bumpbot · Answer

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In a set of twenty numbers, 19 of the 20 numbers are between 40 and 50

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