Sampling Frame vs Population

Three groups sit inside one another and the frame is the middle one. The population is who the question is about. The frame is the list, register, or map the method actually draws from. The sample is who gets selected out of the frame. Only individuals on the frame have any chance of being selected at all, so every probability statement a survey makes is a statement about the frame.

A town has 9,000 adult residents and the pollster works from a landline directory covering 6,300 of them, or 70 percent. The other 2,700 have selection probability zero. Suppose 55 percent of listed adults support a measure and 30 percent of unlisted adults do. The population value is $\frac{0.55(6300) + 0.30(2700)}{9000} = 0.475$, while a flawless random sample of the frame centers on 0.55. The method is off by 0.075 before anyone picks up a phone, and calling 6,300 people rather than 300 does not move it. A census of the frame is still not a census of the population.

The wrong sentence is "the sampling frame is the group the researcher wants to study." That is the population. The frame is the operational stand-in for it, and naming the gap between the two is the entire reason the term exists. A second slip: a frame can be too big as well as too small. A voter list still holding people who moved away, or a mailing list where some households appear twice, contains units outside the population and gives some units two chances of selection.

An SRS drawn from a frame is a simple random sample of the frame. It inherits the frame's gaps exactly, and the absence of the missing group is undercoverage, which no sample size repairs. The first question worth putting to any survey is who the method could never have reached, and the frame answers it rather than the data.

Potential problems with sampling is topic 1.12 in Unit 1.

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.