# Why doesn't a sample proportion also have a binomial distribution

In a binomial setting, the random variable, X, that gives the number of successes is binomially distributed. The sample proportion can then be calculated as $\frac{X}{n}$ where $n$ is your sample size. My textbook states that

This proportion does not have a binomial distribution

however since $\frac{X}{n}$ is simply a scaled version of a binomially distributed random variable $X$, shouldn't it also have a binomial distribution?

• It has the same list of probability masses, but it does not take integer values. – Stéphane Laurent Jun 15 '15 at 4:58
• @StéphaneLaurent That shouldn't matter though, right? – 1110101001 Jun 15 '15 at 5:05
• @1110101001 you would need to reparameterize the distribution – shadowtalker Jun 15 '15 at 5:09
• @ssdecontrol What is meant by reparameterization? Am I correct in assuming it is changing the values of n and p that characterize the number of trials for which the bernoulli experiment is held and the success probability? If so, doesn't it still mean that X/n is still a binomial distribution, even though it doesn't have the same parameters as X does? – 1110101001 Jun 15 '15 at 5:15
• @1110101001 A discrete distribution is given by 1) its support : the set of values on which it is distributed, 2) the list of probability masses of these values. Your scaled binomial distribution is not a binomial distribution because of 1), but it is isomorphic to the binomial distribution because it has the same list in 2). – Stéphane Laurent Jun 15 '15 at 7:44

As you state, the sample proportion is a scaled binomial (under a few assumptions). But a scaled binomial is not a binomial distribution; a binomial can only take on integer values, for example. Of course, it is very easy to figure out the pmf, cdf, expected value, variance, etc. from what we know of the binomial distribution, which I think is what you're getting at. But if you were to say something like "the sample proportion is a binomial, so the expected value is $np$, as is for all binomials", you would be clearly wrong.
If you wanted to be really technical, if $n$ = 1, then the sample proportion is still a binomial distribution.
• A binomial can only take on integer values -- These integer values are the number of successes for each experiment, right? – 1110101001 Jun 15 '15 at 5:21
• correct: of the $n$ trials, the binomial $X$ is the sum of successes – Cliff AB Jun 15 '15 at 5:24