# Approximating non-integer median for CDF of discrete variable

In my googling, it seems the proper way to find the median of a cdf of a discrete variable is to stick to the discrete values provided, even if you overshoot and end up with an x where P(X <= x) > 0.5. But is there a way to approximate an exact value such that P(X <= ) = 0.5?

A straightforward way would be to use the slope between the two points closest to 0.5 to find X. Would this be considered reasonable? If not, are there other methods that better account for the whole distribution instead of just looking at two points?

Edit: It has been made clear that a non-integer value wouldn't really make sense. However, for some context, I'm looking at how many users have converted after a certain number of free trial appointments, and how that number might be applied across many users. You can see why it feels weird to laypeople to say the median is 5 if P(X <= 4) = 0.49 but P(X <= 5) = 0.90. That's an exaggeration, but the kind of issue I face.

• By a discrete variable, do you mean the values are $0$ and $1$?
– Dave
Jun 14, 2021 at 12:18
• Apologies if I misused the term. I meant opposite of continuous? X can only take on discrete values, often integers Jun 14, 2021 at 12:27
• Think of your question for a Poisson variable. The idea of $1.4$ or $6.7$ events as the median does not make sense to me.
– Dave
Jun 14, 2021 at 12:29
• Yes I see it's not correct to do so. I made an edit in the original post for context though. Knowing this "exact median" value, one might be able to say they're willing to give out 100 appointments to 50 people, assuming 50% of them will convert - as an example Jun 14, 2021 at 12:50
• @Dave If you had a Poisson distribution with parameter $\lambda = \log_e(2)\approx 0.693147$ then the median could be $0$ or $1$ or anywhere inbetween Jun 14, 2021 at 13:05

I do not think that this approach is reasonable. The cdf of a discrete random variable is a step function with jumps at the mass points, and if we would like to find a value $$x$$ such that $$F_X(x) = 0.5$$, in most cases we would not be able to do so. Therefore, the approach that you are proposing is not better or worse than just taking any random number between two closest values.