Sampling Distribution vs Population

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.

The population is fixed by the question, not by the data. Defining one means writing a membership rule precise enough to sort any individual in or out: not "voters" but "the 12,400 people registered to vote in this county on October 1". A number computed from the whole population is a parameter, written $\mu$ (mu) for a mean, $p$ for a proportion, $\sigma$ (sigma) for a standard deviation. A parameter is one fixed number, usually unknown, and it does not move when you take a different sample.

Stay with that county. The parameter is $p$, the proportion of all 12,400 registered voters who plan to vote yes. A random sample of 600 gives $\hat{p} = 0.52$ (p-hat), an estimate of $p$ and not $p$ itself. The population size enters in one place only: the sample is $600/12400 = 0.048$ of the population, under the 10 percent ceiling the standard error formula needs, so $\sqrt{(0.52)(0.48)/600} = 0.020$ can stand as the standard error.

The wrong sentence is short and common: "the population is the 600 voters who were surveyed." Those 600 are the sample. The population is the group the conclusion is about, and it exists whether or not anyone measures it. Two smaller slips travel with it. A population need not be people; it can be 4,000 laptop batteries, or every 20-minute interval in a factory's day. And a population is not everyone, because it stops exactly where the question stops, so this survey says nothing about the state.

One piece of intuition is worth killing off. A bigger population does not need a bigger sample. Precision comes from $n$; apart from that 10 percent check, the population size appears nowhere in $\sqrt{p(1-p)/n}$, so a county of 12,400 and a country of 300 million take about the same sample for the same margin of error.

The population you want and the population your method can reach are different things. The list actually drawn from is the sampling frame, and anyone in the population but off that list has no chance of selection. Random sampling is topic 1.11 in Unit 1.