# How sampling distribution of sample proportion is related to binomial distribution

I am reading a book on introductory statistics. In the book, the author introduces the concept of the "sampling distribution of sample proportion" just after explaining the binomial distribution.

I think I've understood the concept of "sampling distribution" and how to take one.

But I have trouble grasping how "sampling distribution of sample proportion" is related to the binomial distribution.

I am also confused with the terminology used in these contexts. For example, when I studied the binomial distribution, $$p$$ stands for the probability of success on each trail. In the context of "sampling distribution of sample proportion", $$p$$ represents the population proportion.

Are these two same?

I am feeling completely lost at this point.

• under iid sampling, n times the sample proportion would be a binomial count. Mar 14, 2020 at 7:26
• @Glen_b-ReinstateMonica, I am sorry but your words don't make sense to me. Can you please elaborate a little?
– Cody
Mar 14, 2020 at 12:30
• The sample proportion is $\hat{p}=X/n$ where $X$="the number of successes in the n trials" and $n$ is the number of trials. If the outcome of each trial is $B_i$, which takes the value $1$ for a success and $0$ otherwise then $X= B_1+B_2+...+B_n$. If the outcomes -- the $B$'s -- are independent and the population $p$ is the same for all of them (independent, identically distributed, or iid), then $X$ is binomial and $X/n$ (the sample proportion) is simply a constant multiple of that. Mar 15, 2020 at 6:41
• 4.2.1 section "Normal Approximation to the Binomial " of Book "STAT 800: Introduction to Applied Statistics" can give more insight. Aug 25, 2022 at 10:52

The terminology may be the key here: The two $$p$$ you cite are in fact the same (if we regard the population as infinitely large). In this context, one trial is drawing one element from the population and recording whether it has the property of interest or not. So the probability of success $$p$$ (the probability of said element having said property) is the relative frequency of the property in the population.

Let's write $$X$$ for the total number of successes, i.e. how many elements we draw overall that do have the property of interest. We assume the population size is infinite, which is practically justified if it is much larger than the sample size $$n$$. Then there is no difference between drawing with or without replacement, and the probability of success always stays the same $$p$$, regardless of what we have already drawn so far.

So we have a run of $$n$$ independent trials, each with a probability of success $$p$$, and we are interested in the total number of successes $$X$$. By definition, this means that $$X$$ has a binomial distribution with parameters $$n$$ and $$p$$.

Now the sample proportion is $$X/n$$, so it differs from $$X$$ only by the constant (non-random) scaling factor $$1/n$$, and therefore the shape of its distribution is the same as the distribution of $$X$$, i.e. a binomial distribution.

• can you shed some more light on how proportion and binomial distribution are related and why binomial distribution is considered as the population?
– Cody
Mar 14, 2020 at 6:43