Sampling Distribution vs Sample

Three different distributions get called "the distribution" and only one of them is this one. The population distribution holds a value for every individual. The distribution of a sample holds the values in the one sample you actually collected. A sampling distribution is neither: its individuals are entire samples of a fixed size $n$, and the number recorded for each is a statistic computed from it, such as $\bar{x}$ (x-bar) or $\hat{p}$ (p-hat).

Small cases can be written out in full. Take the population 2, 4, 6, 8, with $\mu = 5$ (mu) and $\sigma = 2.2361$ (sigma), and draw samples of size 2 with replacement. There are exactly 16 such samples, and their means come out as 2 once, 3 twice, 4 three times, 5 four times, 6 three times, 7 twice, and 8 once. That list is the sampling distribution of $\bar{x}$. Its mean is 5, matching $\mu$, and its standard deviation is 1.5811, which is $\sigma/\sqrt{2}$. The population is flat; the sampling distribution is already triangular at $n = 2$.

The misreading is blunt: "the sampling distribution is the distribution of my sample." A histogram of the 40 numbers you collected is a picture of the sample. As $n$ grows it comes to look more like the population, skew and all, and it does not narrow. The sampling distribution is the thing that narrows, and $\sigma/\sqrt{n}$ describes it rather than your data.

Outside enumerable toy cases you never actually build one. Theory stands in for it, the central limit theorem for means and the binomial for counts, or a simulation approximates it: 10,000 simulated sample means draw a close picture of the sampling distribution without being it.

Topic 2.12 introduces sampling distributions. Every confidence interval and every p-value later in the course is a statement read off one.

One sampling distribution gets reported on the news every month. The US unemployment rate is not a census, it is an estimate from a household survey of about sixty thousand homes, which is why it is revised and why it carries a margin of error at all: the unemployment rate.

A sample is the set of individuals you actually measure, drawn from the larger population you want to describe. Its size is written $n$. A number computed from a sample is a statistic, such as the sample mean $\bar{x}$ (x-bar) or the sample proportion $\hat{p}$ (p-hat), while the matching fact about the population is a parameter, written $\mu$ (mu) or $p$. The sample is the group of individuals, not the number you compute from them.

Poll 1,000 randomly selected voters and suppose 520 say yes. Then $\hat{p} = 520/1000 = 0.52$, an estimate of the unknown $p$. A different random 1,000 would not land on 0.52 again: the standard error is $\sqrt{\frac{0.52 \times 0.48}{1000}} = 0.0158$, so a repeat poll coming back at 0.50 or 0.54 would be ordinary rather than a sign that something went wrong.

Notice what is absent from that calculation. The population size never enters it, which is where the usual objection fails. "A poll of 1,000 cannot speak for 250 million people" sounds like arithmetic, but precision is governed by $n$, not by the fraction of the population sampled. Those 1,000 voters give the same 0.0158 whether the electorate is 250 million or 250 thousand, provided the sample is a small share of it.

What the formula does assume is that chance chose the sample. None of the above survives a sample that was not random: a self-selected sample of 250,000 has a standard error you can still compute and a bias you cannot, which is the point of does a bigger sample fix bias.

Measuring every member of the population instead is a census, and it returns parameters rather than estimates. Sampling exists because most populations are too large, too expensive, or too scattered for that, and because a careful sample often produces better data than a rushed attempt at everyone.