Can a discrtete probability distribution have two medians?

[humy]

I long thought I knew the answer to this but then I keep seeing websites that seem to give me completely contradictory answers to this and now I am totally confused.
Lets say we have some discrtete probability distribution where the random variable of that distribution is the natural number variable x.
Lets say we have;
CDF(x=3) = 0.5
i.e. for cumulative distribution function x=3 input the output of the cumulative distribution function is EXACTLY 0.5.
Would that mean we just very simply have only;
median = 3
or would that mean we have;
median = 3 and 4
i.e. would it mean it has TWO medians?
or would that mean we have;
median = 3.5
i.e. the average of the two values (of 3 and 4)?
Or would that mean we have;
median = [3, 4]
i.e. the median being a real number interval from 3 to 4?

 

[Romsek]

The median in the case of a discrete distribution with an even number of points in the support will be the average of the center two values.

I.e. in this case 3.5

 

[humy]

The median in the case of a discrete distribution with an even number of points in the support will be the average of the center two values.

I.e. in this case 3.5

Thanks.
That's all I wonted to know.
Not that I would think it likely you are mistaken, because I don't, but, Just in case there would be some professional disagreement here, I will wait for at least one other poster to confirm that.
I am writing some maths to be published where I need to explain this aspect of median and it would definitely mar my good work if I got even just one of my facts just a bit wrong so I need to make absolutely sure.

 

[Romsek]

LMGTFY

 

[Dr.Peterson]

Technically, the median of a discrete distribution is not unique: https://en.wikipedia.org/wiki/Median#Probability_distributions

For any probability distribution on the real line R with cumulative distribution function F, regardless of whether it is any kind of continuous probability distribution, in particular an absolutely continuous distribution (which has a probability density function), or a discrete probability distribution, a median is by definition any real number m that satisfies the inequalities​

[MATH]\operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}[/MATH]​

... Any probability distribution on R has at least one median, but in specific cases there may be more than one median. Specifically, if a probability density is zero on an interval [a, b], and the cumulative distribution function at a is 1/2, any value between a and b will also be a median.​

In your example, given that CDF(3) = 0.5, we know that [MATH]\operatorname {P} (X\leq 3) = 0.5[/MATH]. Assuming that CDF(2) < 0.5 and CDF(4) > 0.5, this means that the median can be taken to be any number from 3 to 4 (inclusive).

Commonly we take as median the average of 3 and 4, namely 3.5. But it is better to say that 3.5 is a median, not the median.

 

[humy]

... Any probability distribution on R has at least one median, but in specific cases there may be more than one median. Specifically, if a probability density is zero on an interval [a, b], and the cumulative distribution function at a is 1/2, any value between a and b will also be a median.​

Commonly we take as median the average of 3 and 4, namely 3.5. But it is better to say that 3.5 is a median, not the median.

Thanks.
So I think you imply for my OP example we actually have;
median = [3, 4]
BUT, sometimes for the greater convenience of just stating it as only one single unique median value, we may state the median for my OP example as being;
median = 3.5
But that above is actually stating only one of the medians.

 

[humy]

I STILL see a problem here because I keep seeing websites that seem to give me completely contradictory answers to this.
For example, it says on this wiki website

Binomial distribution - Wikipedia

en.wikipedia.org

that the median of a binomial distribution is in fact;

⌊np⌋ or ⌈np⌉

Please will somebody correct me if I am wrong here but that "or" above implies to me there can be two medians but no more than two medians to this discrete probability distribution.
That in turn implies the the median for my OP example is actually;

median = 3 or 4

and NOT median = [3, 4] or median = 3.5.

Anyone?

 

[humy]

⌊np⌋ or ⌈np⌉

ARR wait a moment!
I think I might have finally figured out where I have gone wrong here!
Is that "or" in that above wiki quote necessarily supposed to be 'exclusive-or' as opposed to 'inclusive-or'?
If so, then wiki has NOT contradicted itself here (about the median) and thus there is no problem here and everything is fine.

 

[Dr.Peterson]

I STILL see a problem here because I keep seeing websites that seem to give me completely contradictory answers to this.
For example, it says on this wiki website

Binomial distribution - Wikipedia

en.wikipedia.org

that the median of a binomial distribution is in fact;

⌊np⌋ or ⌈np⌉

Please will somebody correct me if I am wrong here but that "or" above implies to me there can be two medians but no more than two medians to this discrete probability distribution.
That in turn implies the the median for my OP example is actually;

median = 3 or 4

and NOT median = [3, 4] or median = 3.5.

Anyone?

Did you see that that article gives two different answers, so that it does, in fact, contradict itself?

In the box near the top, as you quoted, it says

Median [MATH]\lfloor np\rfloor[/MATH] or [MATH]\lceil np\rceil[/MATH]​

But down in the section on median, https://en.wikipedia.org/wiki/Binomial_distribution#Median, it says much more fully (my bold):

In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:​

• If np is an integer, then the mean, median, and mode coincide and equal np.​
• Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉.​
• A median m cannot lie too far away from the mean: |m − np| ≤ min{ ln 2, max{p, 1 − p} }.​
• The median is unique and equal to m = round(np) when |m − np| ≤ min{p, 1 − p} (except for the case when p = 1/2 and n is odd).​
• When p = 1/2 and n is odd, any number m in the interval 1/2(n − 1) ≤ m ≤ 1/2(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median.​

I'd say the first is an error; but it could be considered a mere oversimplification, as I would have expected in that location.

So I think you imply for my OP example we actually have;
median = [3, 4]
BUT, sometimes for the greater convenience of just stating it as only one single unique median value, we may state the median for my OP example as being;
median = 3.5
But that above is actually stating only one of the medians.

That's a fair way to say it, except that I wouldn't say the median is the interval [3,4] itself; rather, by the definition, any number in that interval is a median. A median has to be a single number; we just have infinitely many candidates. And 3.5 is the easiest median to talk about. (It also agrees with the traditional way to identify "the" median of a set of numbers, as taught at the elementary level. I discussed the subtleties involved in clearly defining that median in my blog, here.)

 

[humy]

That's a fair way to say it, except that I wouldn't say the median is the interval [3,4] itself; rather, by the definition, any number in that interval is a median.

So perhaps a better way to express that is with;

medians ∈ [3, 4]

and note the use of the plural of 'median' there with 'medians'.

 

[Dr.Peterson]

Or, to avoid confusing people who expect to have one number called the median, you can just say "the median is between 3 and 4".

A lot depends on your context and purpose -- whom you are talking to, and why.