When is the mean = median?

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.

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.?

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.

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

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

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

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

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.

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

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 ?

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

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.

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

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

*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

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.