How do I visualize a sample proportion and its standard error?

I'm studying sampling distributions and confidence intervals for proportions and all is well. However, I don't understand what the standard error of a sample proportion is about. I know about the standard deviation of a sampling distribution, but not the standard error of a sample proportion. If we draw a sample and get a proportion, what does the standard error do for us? I know we can calculate margins of errors with it to construct confidence intervals, but why would a sampled proportion deviate? Am I missing something?

I could easily draw a sampling distribution model, but what about a sample proportion?

• The sample proportion is itself a sample mean. 2 out of 10 people smoke in your first sample; score 1 for smoker and 0 for not and the proportion is the mean of that score 2/10 = 0.2. 3 out of 10 people smoke in the next sample and the sample proportion is 0.3, and so forth. What's different about proportions is largely that the relationship of sample variability and sample mean is quadratic. If the sample proportion is always 0, then it doesn't vary and similarly with 1; so the extreme possibilities imply zero variation. – Nick Cox Jan 22 '17 at 13:06
• But when we calculate the standard error for that proportion of smokers, what is that telling us? I mean; we know that our proportion is 0.2. What does that standard error tell us? sqrt((0.2 * 0.8) / 10) gives us a standard error of approx. 0.126 but what does this number tell me? My sample proportion won't change. – Zimano Jan 22 '17 at 13:12
• You know what your sample proportion is. But the whole point of sampling is to learn about the population, and we also know that there is sampling variation, i.e. different samples won't all agree. That is what we are quantifying. Although the details differ, the sample proportion is like any other sample mean and subject to this kind of uncertainty – Nick Cox Jan 22 '17 at 13:15
• So all the standard error of the sample proportion does is quantify the sampling variation? What happens if we happen to be "unlucky" with a sample and get p = 0.1 and q = 0.9. The standard error produced by those numbers is wildly different than, say, p = 0.4 and q = 0.6. How can it quantify something so dependent on a per-sample basis! – Zimano Jan 22 '17 at 13:43
• No; the standard error is not any individual difference or measurement for any individual sample proportion. The question is like saying what's the use of 50 mm standard deviation of people's heights if there are giants and dwarfs? The standard error is just a summary of variability, no less, no more. Some differences will be greater, some smaller. – Nick Cox Jan 22 '17 at 15:02