The CLT for Sums and Proportions

A sum is simply all the values added up (the total weight of 64 suitcases), and a sample proportion is the fraction of a sample with some trait (40% of those polled said yes). The point of this lesson is that the last lesson's bell curve covers both, which is how airlines check payloads and pollsters get their margins of error. The two formulas look like a lot, but each has just two parts: a center and a spread. For a total, the center is $n\mu$ ($n$ values, each about $\mu$) and the spread is $\sqrt n\,\sigma$ (it grows, but only like a square root). For a proportion, the center is $p$ (the true rate) and the spread is $\sqrt{\frac{p(1 - p)}{n}}$ (which shrinks as the sample grows). Both come straight from the CLT you just saw, so there's less new here than it looks.

Totals vs averages. The total of $n$ values is $n$ times their average, so you get its distribution by stretching the average's distribution by a factor of $n$. Its mean is $n\mu$ and its standard deviation is $n\cdot\frac{\sigma}{\sqrt n} = \sqrt n\,\sigma$. Notice that the spread of the total grows, while the spread of the average shrinks. Both are true at once: the total wanders more in absolute terms, but less relative to its size.

Proportions. Here's the trick that makes this work. Record each individual as a 1 (success) or 0. Each one has mean $p$ and standard deviation $\sqrt{p(1 - p)}$. Now average $n$ of them: the average of a bunch of 0s and 1s is exactly the fraction of 1s, which is $\hat p$. So $\hat p$ is a sample mean in disguise, and the CLT applies with $\sigma = \sqrt{p(1 - p)}$. Divide by $\sqrt n$ and you have its standard error.

When is the approximation good? For proportions, check that $np \ge 10$ and $n(1 - p) \ge 10$. This is also the normal approximation to the binomial, since the count of successes is a binomial total.

A common mistake. It's tempting to think the total of 64 suitcases has standard deviation $64 \times 10 = 640$. It doesn't, because independent ups and downs partly cancel. Variances add, not standard deviations, so the variance is $64 \times 10^2$ and the standard deviation is $\sqrt{64} \times 10 = 80$.

Airlines check total payloads, insurers model total claims, and every opinion poll's margin of error comes from the normal approximation for proportions.