Is the standard deviation of numbers x, y, and z, positive?

Question

GMAT CLUB TESTS' FRESH QUESTION

(1) The average (arithmetic mean) of x, y, and z, is less than x
(2) The median of x, y, and z, is greater than z

M36-33

Bunuel · Accepted Answer

Official Solution:

Is the standard deviation of numbers  x ,  y , and  z , positive? 
 
The standard deviation of a list shows how much variation there is from the mean, how widespread a given set is. So, a low standard deviation indicates that the data points tend to be very close to the mean, whereas a high standard deviation indicates that the data are spread out over a large range of values. So, basically we can say that it in a sense measures the distance and the distance cannot be negative, which means that  the standard deviation of any list is greater than or equal to zero :  SD\geq0 .
 
Next, the standard deviation of a list is zero if and only the set consists of identical numbers (or which is the same if the list consists of only one number).
 
So, the question basically asks whether  x=y=z ., If  x=y=z , then the standard deviation will be zero and if not, then the standard deviation will be positive.
 
(1) The average (arithmetic mean) of  x ,  y , and  z , is less than  x 
 
If  x=y=z  were true, then the average of  x ,  y , and  z  would be  x  and since we are told that the average is less than  x , then  x=y=z  is not true and thus the standard deviation is positive. Sufficient.
 
(2) The median of  x ,  y , and  z , is greater than  z 
 
If  x=y=z  were true, then the median of  x ,  y , and  z  would be  z  and since we are told that the median is greater than  z , then  x=y=z  is not true and thus the standard deviation is positive. Sufficient.

Answer: D

chetan2u · Answer

the standard deviation can be positive or 0, never negative. It will be 0 only when each element is same

Is the standard deviation of numbers x, y, and z, positive?

(1) The average (arithmetic mean) of x, y, and z, is less than x
 this means the elements are not the same, so there is some dispersion and it is always positive 
Sufficient
(2) The median of x, y, and z, is greater than z
Same again. All elements are not the same.
SD is positive
Sufficient

D

GmatDaddy · Answer

The SD will always be Greater Than or Equal to 0. It can never be Negative.

{ Background Information: 
Case 1: SD=0
This means that all the elements of the set are the same.

Case 2: SD>0
This means that the all the elements of the set are not equal.}

Getting back to the Question:
Statement 1:(1) The average (arithmetic mean) of x, y, and z, is less than x
This means that atleast 1 element of the 3 is different than the rest 2.
Therefore, the SD >0.

Statement 2: The median of x, y, and z, is greater than z
This means that at least z is different than x or y
This implies that atleast on e element is different in the set.
Therefore, SD >0.

Hence,  the answer is D .

Please Note  that the probability of GMAT asking test takers to calculate the value of SD is very low. However, it does asks questions on the basics of SD.
This is a very good question testing the basics of GMAT.

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

Trick here is to remember that standard deviation of a set can only be positive or 0. Standard deviation will be 0 if all elements are the same.

1) Given that average of x, y and z is less than x means that there is at least one number that is less than x. This means that all elements are not equal. Hence, standard deviation is not 0. Thus it has to be positive.  Hence, sufficient

2) Given that median of the set is greater than z means that z is the smallest number and that it is different from the other two numbers. The latter is true because if any of the other two numbers was equal to z, median would have been equal to z, which is not the case. Thus, all elements are not equal. Hence, standard deviation is not 0. Thus it has to be positive.  Hence, sufficient

Hence, answer is D

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Is the standard deviation of numbers x, y, and z, positive?

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