Lesson: Mean, Median & Mode

Comment on Mean, Median & Mode

can you please explain the solution of the below? Good answer explanations not given by users of GMAT Club

What is the average of x, y, and z?
(1) 2x + y + 4z = 23
(2) 3x + 4y + z = 22

Sir, is there any easy way to solve the problem

For the first 3 months of last year, the average daily rainfall in Lancaster was 6cm. For last year as a whole, the average daily rainfall was 12cm. What was the average daily rainfall for the last 9 months of last year?
gmat-admin's picture

So many practice questions on this topic. Is it considered a major part in the actual GMAT test?
gmat-admin's picture

Statistics questions (especially questions about mean and median) are VERY COMMON on the GMAT.
Of course, you need not answer every question above, but be sure to understand the various types of questions the test-maker can create.

I think this question has 2 answers that would work: C and E. If you use the multiple 3 it gives 5 as the median and when you use multiple 6 it gives 8 as the median which is also listed.

If x is a positive, single-digit integer such that 4/3*x, 2x, x, and x + 2, and 3x – 2 form a non-ordered list of consecutive integers, which of the following could be the median of that list?

A. 3
B. 4
C. 5
D. 6
E. 8
gmat-admin's picture

That's great that you recognized that x must be a multiple of 3. However, x = 6 does not yield five CONSECUTIVE integers.

If x = 6, then 4x/3 = 8, 2x = 12, x = 6, x + 2 = 8 and 3x - 2 = 14. So, the five numbers (in ascending order) are: 6, 8, 8, 12, 14 (since these aren't consecutive integers, x cannot equal 6.


Guess I missed that small detail. Whoops!
gmat-admin's picture

It happens to everyone :-)

Hi Brent,

The final answer of "A" in the answer box (to the problem below) does not match your simplified solution at the end of your problem solving.

The average (arithmetic mean) of y numbers is x. If z is added to the numbers, the new average (arithmetic mean) will be z-5. What is the value of z in terms of x and y?
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-average-arithmetic-mean-of-y-numbers-is-x...

Good catch - thanks!
I've edited my response accordingly.


Hi Brent,
The problem below gives me some difficulty when dealing with how to solve/understand for the Absolute Value portion. In particular, I don't quite understand how Bunuel is manipulating the absolute values or what it is I'm not understanding.

A set S = {x, -8, -5, -4, 4, 6, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?

(A) -1
(B) 0
(C) 1
(D) 2
(E) Cannot be determined
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-set-s-x-8-5-4-4-6-9-y-with-elements-arrang-...

Nice question!

First off, the question tells us that the numbers are arranged in ascending order.
So, we know that x ≤ -8, and y ≥ 9

There are 8 elements in the set. So, the median = the average of the two middlemost values.
Here, the two middlemost values are -4 and 4
So, the median = (-4 + 4)/2 = 0/2 = 0

Since the median and the mean of the set are EQUAL, we know that the mean is also 0

That is, [x + (-8) + (-5) + (-4) + 4 + 6 + 9 + y]/8 = 0
Multiply both sides by 8 to get: x + (-8) + (-5) + (-4) + 4 + 6 + 9 + y = 0
Simplify: x + y + 2 = 0
This means x + y = -2

So, here's what we know:
x + y = -2
x ≤ -8
y ≥ 9

Let's find some values of x and y and see where this leads us....

x = -12 and y = 10
In this case, |x|-|y|= |-12|-|10| = 12 - 10 = 2

x = -13 and y = 11
In this case, |x|-|y|= |-13|-|11| = 13 - 11 = 2

x = -12.5 and y = 10.5
In this case, |x|-|y|= |-12.5|-|10.5| = 12.5 - 10.5 = 2

x = -100 and y = 98
In this case, |x|-|y|= |-100|-|98| = 100 - 98 = 2

As we can see, the answer will always be 2

Answer: D


Thank you Brent! Great way to break it apart without having to worry about the complexity of dealing with Absolute value. However, I'm still bugged (mostly frustrated) by one of the methods posted by "Gmatify" (See below). If you could walk me through it, it would really help me to see this problem both ways.
Gmatify's description and method of dealing with Absolute value seems off to me, especially the bit about, "the modulus of a negative number opens with a negative sign." and the rest of the way he solves the equation seems "off!" What am I not seeing here??

Opening the modulus with appropriate signs:
Modulus of any number is the absolute value of the number, or simply the positive value
Always remember that the modulus of a negative number opens with a negative sign and of a positive number opens with a positive sign

We have ,
|x| - |y| = -x - y = -(x + y) = -(-2) = 2
Hence Option D
gmat-admin's picture

Gmatify is trying to explain how/why he/she wrote: |x| - |y| = -x - y

Before stating a rule about absolute values, let's look at some examples of the absolute value of a NEGATIVE number.
|-3| = 3
|-8| = 8
|-11| = 11
|-1.2| = 1.2

In general, if x is NEGATIVE, then |x| = -x
For example, if x = -6, then |x| = |-6| = 6 = -(-6) = -x

Now, let's look at some examples of the absolute value of a POSITIVE number.
|3| = 3
|9| = 9
|5.2| = 5.2
|19| = 19

In general, if y is POSITIVE, then |y| = y
For example, if y = 7, then |y| = |7| = 7 = y

In the original question, we're told (indirectly) that x is NEGATIVE and y is POSITIVE
So, it must be the case that |x| = -x and |y| = y

So, |x| - |y| = -x - y = -(x + y)

Does that help?


Hi Brent,

I need a little clarification on the challenge problem posted below.

Here is my question:

How do you figure that “3” is the median of set “A” when “J” is unknown? It just assumes J will somehow be negative, but what if it is positive?

Set A consists of integers -9, 8, 3, 10, and J; Set B consists of integers -2, 5, 0, 7, -6, and T. If R is the median of Set A and W is the mode of set B, and R^W is a factor of 34, what is the value of T if J is negative?
(E) 5
gmat-admin's picture

Question link: http://www.veritasprep.com/blog/2010/10/gmat-challenge-question-solve-this/

The answer to your question ("How do you figure that 3 is the median of set A when J is unknown?) lies at the very end of the question stem, where it says "......what is the value of T if J is NEGATIVE?"

So, even though we don't know the actual value of J, we do know that J is some NEGATIVE number.

To determine the median of set A, we must arrange the values in ASCENDING order.

We know that J is NEGATIVE, and we also have another negative value (-9) in set A.

This gives us two possible cases: J < -9 OR -9 < J

It turns out that both cases yield the SAME MEDIAN for set A.

Case i: J < -9
When we arrange set A in ASCENDING order, we get: {J, -9, 3, 8, 10}. The MEDIAN = 3

Case ii: -9 < J
When we arrange set A in ASCENDING order, we get: {-9, J, 3, 8, 10}. The MEDIAN = 3

In both cases, the median is 3.

In other words, R = 3

Does that help?


Well, I guess that was right in front of my face, but with so many variables I suppose that's how they "get you." However, now my question shifts to how do we find "T
" now?

gmat-admin's picture

Set B:{-2, 5, 0, 7, -6, T}
W is the mode of set B

The key word here is "THE."
This means there is ONLY ONE mode.
So, for example, if T = 7, then set B is {-2, 5, 0, 7, -6, 7}, in which case the mode is 7
Conversely, if T = 11, then set B is {-2, 5, 0, 7, -6, 11}, in which case the modes are -2, 5, 0, 7, -6 and 11

Since the key word "THE" tells us that there is ONLY ONE mode, we know that T is one of the 5 values: -2, 5, 0, 7 or -6

We're also told that R^W is a factor of 34

We know (from my post above) that R = 3
So how can 3^W be a factor of 34, when the factors of 34 are: 1, 2, 17 and 34?

We know that, if W is an integer, 3^W CANNOT equal 2, 17 or 34

HOWEVER, 3^W CAN equal 1, when W = 0

So, it MUST be the case that W = 0

If W = 0, then 0 is the mode of the set {-2, 5, 0, 7, -6, T}

If 0 is THE mode of the set {-2, 5, 0, 7, -6, T}, then it must be the case that T = 0

Answer: B


CLICKED! Thank you so much for taking me through that. That has got to be a 800 level question with that many twists and turns. Wow! See why you saved it for last.
gmat-admin's picture

Yes, it's definitely a tough one!

What if w=-2? 3^-2 is equal to 1/9
(2*17)/(1/9)=34*9=306 which is an integer. So W could equal -2 also.

What is wrong with this?
gmat-admin's picture

Hi Kaori,

Which question are you referring to? Please provide a link.


This the link. Question link: http://www.veritasprep.com/blog/2010/10/gmat-challenge-question-solve-this/

I was talking about R^W
gmat-admin's picture

Link: http://www.veritasprep.com/blog/2010/10/gmat-challenge-question-solve-this/

In your suggested case (r = 3 and w = -2), r^w = 1/9
However, the question tells us that r^w is a factor of 34, and 1/9 is NOT a factor of 34.

Important: The factors (aka divisors) of a certain integer are always integers.

So, for example, the positive factors of 34 are: 1, 2, 17 and 34


Great. I was confused as I thought a factor is any number that gives an integer when divided by.

Factors can't be fractions and should be integers.
Things are clear now!

Set Q consists of 6 consecutive even integers beginning with -4, and Set P consists of 4 consecutive odd integers beginning with 1. If Set M consists of all numbers from both Set P and Set Q, how much greater is the median of Set M than the mean of Set M?

A. 3
B. 2.5
C. 0.8
D. 0.3
E. 0.25

Hi Brent, I feel the question here is incomplete.

Set Q can have two possibilities: -4, -6, -8....... or -4, -2, 0......

Similarly Set P will have two possibilities: 1,3,5,7 or 1,-1,-3,-5

Am I understanding this correctly?
gmat-admin's picture

Question link: https://gmatclub.com/forum/set-q-consists-of-6-consecutive-even-integers...

Perhaps the question could/should be worded differently to avoid ambiguity.

"beginning with" assumes we're listing the values in ascending order. So, we get: -4, -2, 0, 2, etc.

Perhaps the question should read as follows:
"Set Q consists of 6 consecutive even integers, WITH -4 AS THE SMALLEST (LEAST) INTEGER."


sir how to approach this question??
gmat-admin's picture


Just so I understand this right, as long as there are different numbers in a set the relationship between the median and the mean could be as follows:

Numbers below the average lie below the median. Similarly, numbers above the median lie above the average.

That seems to be the logic used by almost all the people in this thread.
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-certain-list-consists-of-400-different-numb...

I don't agree with your statement that "numbers below the average lie below the median."

Consider the set {1, 2, 3, 4, 5, 6, 119}
The median = 4
The average = 20
Here, the numbers 5 and 6 are less than the average, but they are greater than the median.

You can also see that the above set contradicts your suggestion that "numbers above the median lie above the average."
That is, numbers 5 and 6 are greater than the median, but they are not greater than the average.


If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?

(A) (Q - 1)/2 + 120
(B) Q/2 + 119
(C) Q/2 + 120
(D) (Q + 119)/2
(E) (Q + 120)/2

please how to solve this one
gmat-admin's picture

We can solve this Variables in the Answer Choices (VIAC) question algebraically or using the INPUT-OUTPUT approach.

Here's my INPUT-OUTPUT approach: https://gmatclub.com/forum/if-q-is-an-odd-number-and-the-median-of-q-con...


Hey Brent,

I have a question regarding this Q:


How could we summarize the answer in a rule?

Is it like: If each value of the mean is affected by a certain percent increase, we don´t know the total amount if we don´t know each salary? Or how could we sum this up in an understandable and easily applicable rule?


gmat-admin's picture

Question link: https://gmatclub.com/forum/last-year-the-average-arithmetic-mean-salary-...

Hmmm, I'm not sure we can create a useful rule from this, other than something like:
If each salary increases by x percent, then the average of the salaries must also increase by x percent.

Hello Sir,

I have come across this question, and I just have one simple question.

The question says, "he average of a set of five distinct integers is 300. If each number is less than 2,000, and the median of the set is the greatest possible value, what is the sum of the two smallest numbers?"

- The part I don't get is if the question says that "the median of the set is the greatest possible value," then how can the median be 1997 because if that is the case, 1998 and 1999 are bigger than 1997?

Am I misinterpreting the question?

-Thank you
gmat-admin's picture

Question link: https://gmatclub.com/forum/the-average-of-a-set-of-five-distinct-integer...

When all 5 integers are arranged in ASCENDING ORDER, the median will be the MIDDLE value.
So, for example, if the 5 numbers are a, b, c, d, and e, and if a < b < c < d < e, then the median is c.

Since the integers are DISTINCT and less than 2,000, the greatest possible value is 1999
In other words, e = 1999

The next biggest possible value is 1998.
So, d = 1998

The next biggest possible value is 1997.
So, c = 1997

So, 1997 is the greatest possible value of the median.

Does that help?



Hi Brent,

In this problem, isnt it not possible for integer g in set P to be 0 ? wouldnt that make range of P and range of Q equal to one another ?

Am I missing something here ?

gmat-admin's picture

Question link: https://gmatclub.com/forum/if-a-b-c-d-e-f-and-g-are-distinct-integers-wh...

If g = 0, the ranges may or may not be equal

Q = {1, 2, 3, 4, 5, 6}
If g = 0, we get: P = {0, 1, 2, 3, 4, 5, 6}
In this case, range P ≥ range Q.

Q = {-3, -2, -1, 1, 2, 3}
If g = 0, we get: P = {-3, -2, -1, 0, 1, 2, 3}
In this case, range P = range Q.

This is in line with answer choice A (the correct answer), which says "Range P ≥ Range Q"

Does that help?


Yes, thanks !

Hi Brent,

Help with this one please ?


gmat-admin's picture

Hi Brent, in below Q why are we assuming that numbers would be positive only? If Negative no’s are allowed, then answers should be E. Right?
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-certain-list-consists-of-400-different-numb...

We don't need to assume all of the numbers are positive to correctly answer this question.
If we allow some (or even all) how the numbers to be negative, the correct answer is still D.

I hope that helps.
Cheers, Brent

Hi Brent!

Reference is OG 2019, question 392.

Could you pls explain why answer is C instead of E? Is it not possible that there are exactly 0 projects where there are exactly 3 employees assigned per project?

gmat-admin's picture

Hi Zargam.
In the future, please either state the question, or provide a link to the question on GMAT Club (otherwise, I have to search my office for a copy of the OG2019 :-)

Question link: https://gmatclub.com/forum/what-is-the-median-number-of-employees-assign...

No it isn't possible that 0 projects have exactly 3 employees assigned.

Statement 2 tells us that 35% of the projects have 0, 1 or 2 employees assigned to them.
Statement 1 tells us that 25% of the projects have 4, 5, 6, 7, ..... etc employees assigned to them.

At this point, we have accounted for projects that have 0, 1, 2, 4, 5, 6, 7.... etc employees assigned to them.
Also note that we have accounted for 60% of all projects.
This means the remaining 40% of the projects must have exactly 3 employees.

Alternatively, if you try to find to counter-examples that satisfy both statements, you'll find that it is impossible to do so.

Does that help?

Sorry for the trouble Brent!

I am in fact working with a hard copy of OG2019 :p Will write the question next time.
gmat-admin's picture

Thanks Zargam!


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