When is the mean = median?

Question

From my understanding, mean = median if, 
1. the set consists of evenly spaced numbers 
2. if all the members of the set are equal 
3. set has just one number

Is there any thing else? In DS type of "is mean = median" questions, what do you have to know to be sure that mean = median?

GMATPill · Accepted Answer

Perhaps the easiest way to understand when mean = median is this one reqiurement:

1) When the distribution above the mean is symmetrical to the distribution below the mean.

So think of the middle point. Whatever pattern of data you have above the middle point is mirrored on the other side (below) the middle point.

When this happens, the mean will effectively be the same as the median (the middle number by rank).

This is satisfied when data is evenly spaced out. It's also true in other mirror-like cases.

Sure the distribution does not necessarily have to be exactly mirrored on both sides. But as long as they roughly cancel each other out, then you'll have a situation where the mean = median.

Cares · Answer

Not true, here's an example where mean = median and none of your criteria listed are met.

{0,50,50,50,100}

For any evenly spaced set, the mean of the set is always equal to the median. But not vice versa!

cyberjadugar · Answer

Hi,

If a series is an  arithmetic progression then mean value will always be equal to median , but vice versa is not true.

Thus, any series can be formed other than the ones you have mentioned where mean = median.

bellcurve · Answer

Thanks guys!

I did not say that the list that I provided is the exhaustive list. I just wanted add more to it. So if the Q is "is mean=median" What constitute sufficiency other than arithmetic progression and/or evenly spaced sets? Anything about the relationship that can be expressed in terms of range, relationship between the smallest or largest number, or the relationship between median and other numbers etc.?

OldFritz · Answer

Median=Mean when:
1) Numbers are in a sequence i.e. evenly spaced (and that includes consecutive numbers)
2) Only one item in the set
3) All members of the set are equal

Cheers.

geometric · Answer

I've always thought of it like this, if you take the average of a group of evenly spaced numbers and that result is the middle number, then voila, mean = median.

For instance,

10, 20, 30, 50, 70, 80, 90

Mean = 50
Median = 50

heintzst · Answer

When there's no skew in the distribution.....

It's nice if you check out intro statistics (say, AP)

highwyre237 · Answer

Simple explination,

Mean = median
When adding the diffence of every number from the mean in a set together = 0
example 
(1,2,4,6,7) 
Mean = 20/5 = 4

4-1=3
4-2=2
4-4=0
4-6=-2
4-7=-3

3+2+0-2-3=0

gjonrexha · Answer

There are specific rules which allow you to determine when the mean equals the medium but it's better to just mathematically out the set of numbers together. The median will always be the middle value regardless of the set. If it's even amount of numbers it will be the average of the two middle numbers. If you add up all the numbers determine the mean and if it is equal to the middle value. The mean = the median.

avigutman · Answer

The mean equals the median if and only if all the deviations of all the terms from the median cancel out.

Posted from my mobile device

sheldoncooper · Answer

AviGutman  /  GMATPill

In which condition does the vice versa hold true ? ie. Median = Mean ?

Or is it always insufficient to prove that Median = Mean even in a evenly spaced set ?

avigutman · Answer

Any time the mean = median, the median = mean...
Equations are symmetrical in nature.

AG95 · Answer

Adding a 4th line item:  
1. the set consists of evenly spaced numbers
2. if all the members of the set are equal
3. set has just one number
4. In case of consecutive integers. This is special case of evenly spaced numbers.

sujoykrdatta · Answer

There are some specific cases where mean DEFINITELY equals the median:

1. All the terms are equal (or there is only one number)
Ex 1) 2, 2, 2, 2
Ex 2) 4

2. The terms are equally spaced (arithmetic progression)
Ex 1) 1, 3, 5, 7
Ex 2) 1, 2, 3, 4, 5

3. The numbers are symmetric about the mean:
Ex 1) 1, 6, 7, 8, 13
Observe that 1 is 6 less than 7, 13 is 6 more than 7; 6 is 1 less than 7 and 8 is 1 more than 7
Ex 2) 1, 6, 8, 13

4. The random case: 
Ex) 1, 1, 7, 8, 18
Median = 7 and mean = 7; however, there is no pattern
The only way to identify this is the sum of the deviations about the median is zero: 
Deviations about 7 are: 
-6, -6, 0, 1, 11 => observe that their sum is zero

Hope this helps!

Posted from my mobile device

ReedArnoldMPREP · Answer

The average is the median when a set is perfectly symmetrical. This most commonly happens in equally spaced sets (e.g. {4,7,10,13,16}) but it can happen in others. {5,10,31,51,71,92,97} for example, has a median and mean of 51. This video can help explain why this is true (and has a question dealing specifically with): https://www.youtube.com/watch?v=jaeH24VHKBQ

Shubhamdix10 · Answer

*Median=Mean when:
1) Numbers are in AP (cant say anything about std devn)
2) evenly spaced/skewed around median(cant say anything about std devn)
3) Only one item in the set (here std devn = 0)
4) All members of the set are equal (here std devn = 0)

*If a series is an arithmetic progression then mean value will always be equal to median, but vice versa is not true.
ex: (10,15,20,25,30)------------
here, mean=median=20

*For any evenly spaced set i.e. evenly spaced/skewed around median, the mean of the set is always equal to the median. But not vice versa...
i.e. a line through median should divide the graph in two identical mirror reflection-like parts
ex: (1,5,6,66,126,127,131)------
here, mean=median=66

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

When is the mean = median?

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

GMAT Problem Solving (PS) Questions

When is the mean = median?

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