If the variables, X, Y, and Z take on only the values 10, 20, 30, 40

Question

https://gmatclub.com/forum/download/file.php?id=22686 If the variables, X, Y, and Z take on only the values 10, 20, 30, 40, 50, 60, or 70 with frequencies indicated by the shaded regions above, for which of the frequency distributions is the mean equal to the median? (A) X only (B) Y only (C) Z only (D) X and Y (E) X and Z Untitled.png

Bunuel · Accepted Answer

SOLUTION

https://gmatclub.com/forum/download/file.php?id=22686 
 If the variables, X, Y, and Z take on only the values 10, 20, 30, 40, 50, 60, or 70 with frequencies indicated by the shaded regions above, for which of the frequency distributions is the mean equal to the median?

(A) X only
(B) Y only
(C) Z only
(D) X and Y
(E) X and Z

The frequency distributions for both X and Z are symmetric about 40, which means that both X and Z have  mean = median = 40 .

Answer: E.

Eshan · Answer

For Mean and Median to be same, the data should be evenly distributed around the mean. Also, if the distribution is symmetrical, the mean is equal to median.

In X and Z, we can conclude that the mean and median is same because the distribution is symmetrical around the mean (40).

In Y, however, median will be a clearly be multiple of 10 but the mean comes out to be 760/18 which is not a multiple of 10. So clearly Y is not an option to pick.

Therefore, IMO, answer is E.

WoundedTiger · Answer

Untitled.png If the variables, X, Y, and Z take on only the values 10, 20, 30, 40, 50, 60, or 70 with frequencies indicated by the shaded regions above, for which of the frequency distributions is the mean equal to the median? (A) X only (B) Y only (C) Z only (D) X and Y (E) X and Z Sol: Ans is E Consider data for Variable X We see that it is symmetric about 40 and hence the distribution is equally spaced and therefore Mean =Median We can also spread the distribution as 10,20,20,30,30,30,40,40,40,40,50,50,50,60,60,70.. Strike out from extremes and we get Median= 40 And the Mean will be (40*4+10*1+20*2+30*3+50*3+60*2+70*1)/16 or 640/16= 40 Similar symmetry can be observed for Z distribution as well.

sanjoo · Answer

Isnt it too dificult to culculate !

I was going for x only.. bt looking at the z ..it seems it will have same median and mean..what is easiest way to solve

CCMBA · Answer

When the distribution is symmetric, mean equals median. Thus, X and Z must be in the answer. There is no choice with X, Y, and Z, so there is no need to check Y. The answer is E.

riskietech · Answer

For X: The distribution is symmetric and more populated at mean. So for this mean = median
For Y: The distribution is neither symmetric nor populated at mean. So for this mean is not equal to median
For Z: The distribution is symmetric but more populated at ends. So mean is shifted towards right side. Thus mean is not equal to median.

Answer: A

PareshGmat · Answer

Answer = (E) X and Z

Looking at the charts, see that X & Z are symmetric.

So, Answer = E

rhine29388 · Answer

As per the 3 figures given in the question , it can be clearly seen that for graphs X and Z mean is equal to median
correct answer - E

SeregaP · Answer

try to estimate, this is what worked for me: 
both X and Z looks similar and approx. gets us to the same mean and median

sadikabid27 · Answer

Bunuel ,  VeritasPrepKarishma ,  EgmatQuantExpert  please share the theory of these types of problem.

KarishmaB · Answer

The theory of this question is the same as the theory of mean and median. It uses no new concepts. In both X and Z, the numbers are symmetrical about the centre. So the sum of the deviation of numbers on the left of the centre is same as the sum of the deviation of the numbers on the right.
This post explains finding mean using deviations: https://www.gmatclub.com/forum/veritas- ... eviations/

If the set of numbers is 10, 15, 15, 20, 25, 25, 30, the mean will simply be 20. Notice the symmetry around the centre, 20. The median will obviously be 20 too.

The case with X and Z is exactly the same.

Y has a random sequence. There is no symmetry about the centre and hence the mean will not be the same as the median.

KarishmaB · Answer

Responding to a pm:
Go back to the basics. What is mean? It's the average, the single value that can represent all the values. How do you find it? You multiply each value by its frequency, add them all up and divide by the sum of the frequencies.
What is the mean of: 10, 20, 30 - we know it is 20
What is the mean of: 10, 10, 20, 30, 30 - again 20. Why? Because if there is one value '10' which is 10 less than 20, then there is also a value '30' which is 10 more than 20. So effectively, both 10 and 30 give us two 20s.
Similarly, here X is: 10, 20, 20, 30, 30, 30, 40, 40, 40, 40, 50, 50, 50, 60, 60, 70
40 is the mean because we have three 30s and three 50s to balance out. We have two 20s and two 60s to balance out and we have a 10 and a 70 to balance out again.
What is median? Median is the middle value when you arrange the numbers in increasing/decreasing order. We can see that 40 will be the middle value too since there are equal number of total elements on both sides of 40. We have 6 elements smaller than 40 and 6 elements greater than 40. Hence median = 40.

Hence, mean = median for X.

We can reason out the same thing for Z too in exactly the same way.

The only possible answer is (E). I don't have to worry about Y.

MAnkur · Answer

In addition to the above logic mentioned for symmetry around the center that conveys for equal mean and median, one can also quickly analyze to find the mean by using the benchmark value of 40 in X and Z. 40 being a benchmark value in X can easily be seen to get a mean of 40 by calculating the positive and negative deviations o either side. Same logic can be extended to see that the benchmark value of 40 in Z also gives equal positive and negative deviations on either side.
The answer to this question is figure X and Z.

LeenaSai · Answer

What an awsm explanation!

Posted from my mobile device

MHIKER · Answer

The data distributions for X and Z are symmetric about 4. So the mean and median of the  X  and  Z  are equal  =40

The answer is  E

sriharsha4444 · Answer

hi  KarishmaB , Thanks for the great replies across GMAT club. They have been very helpful.

My question: Is it guaranteed that if there isn't a symmetry in the histogram, then mean ≠ median ?

KarishmaB · Answer

No, it is not essential that mean will not be equal to median if there is no symmetry.

Consider: 2, 3, 6, 7, 12

Mean = median = 6

But here the options are such that we can ignore Y.

ManishaPrepMinds · Answer

We are given three frequency distributions for variables X, Y, and Z. We need to find for which distributions the mean is equal to the median.
  Key Idea:  
For the mean to equal the median, the distribution must be  symmetric . In a symmetric distribution, the data is evenly spread around the center, and both sides look like mirror images.
 Analyzing Each Distribution: 
 
  Distribution X: 
 
 The frequencies are evenly spread out on both sides of the center (around 40). 
 This distribution is symmetric, so the mean equals the median.   
  Distribution Y: 
 
 The frequencies are heavier on the left side (more values at 10, 20, 30) and fewer on the right side (at 50, 60, 70). 
 This distribution is  skewed  (not symmetric), so the mean does not equal the median.   
  Distribution Z: 
 
 The frequencies are evenly spread around the center (40), creating a mirror image on both sides. 
 This distribution is also symmetric, so the mean equals the median.    
 Conclusion: 
The mean equals the median for  distributions X and Z  because both are symmetric.

Correct Answer:  (E)

Natansha · Answer

Hi  KarishmaB  following up on this, if our options were not how they are, then we would have to check even the mean & median of Y as well, is that correct?

If the variables, X, Y, and Z take on only the values 10, 20, 30, 40

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If the variables, X, Y, and Z take on only the values 10, 20, 30, 40

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards