Undercoverage vs Convenience Sample

Undercoverage is a gap between two groups that are easy to run together: the population you want to describe, and the sampling frame, the list your method can actually draw from. Anyone in the first but not the second has probability zero of being selected, so those people are missing from every sample the method will ever produce, not just from this one.

A college with 4,000 students wants the proportion who eat at the dining hall at least twice a week, and draws a random sample from the residence hall roster. That roster holds the 2,500 students who live on campus and none of the 1,500 commuters. Suppose 0.80 of residents and 0.25 of commuters eat there that often. The parameter is $\frac{0.80(2500) + 0.25(1500)}{4000} = \frac{2375}{4000} = 0.594$, while the method is aimed at 0.80. Sample after sample scatters around 0.80, about 0.21 above the truth.

"Then survey more students" is the repair people reach for, and it is the wrong one. The frame does not change when the sample grows. Draw 2,000 of the 2,500 residents and the estimate is still centered on 0.80, now with a much smaller standard error, which only makes a wrong answer look authoritative.

The test that separates undercoverage from its neighbors is whether the person could have been selected at all. The commuters could not; they were never on the roster. A commuter who was on the list, was contacted, and never replied would be nonresponse bias instead, a failure one stage later.

One boundary keeps this honest. An incomplete frame biases an estimate only when the missing group differs on the variable being measured. If commuters ate at the dining hall at the same 0.80 rate as residents, this estimate would sit on the parameter even though the roster is just as incomplete, and that same roster would still wreck a survey about parking. Topic 1.12 wants both halves: name the excluded group, then say which way the estimate moves.

A convenience sample is chosen by accessibility: the researcher takes the individuals who are easiest to reach, and no chance mechanism decides who is in. Nobody has a known probability of selection, and the parts of the population that were never within reach have probability zero, so the sample is not random in the sense the inference formulas require.

Suppose a reporter stands outside a gym at 6 a.m., interviews the first 60 people who leave, and finds that 41 of them, or $41/60 \approx 0.68$, exercise daily. A calculator will produce a 95 percent confidence interval for that: $0.68 \pm 1.96\sqrt{\frac{0.68 \times 0.32}{60}}$, which is $0.68 \pm 0.118$, or 0.562 to 0.798. The arithmetic is correct and the interval is worthless, because the randomness condition behind it was never met. It describes 60 people leaving one gym at 6 a.m. and nothing wider.

The sentence that keeps convenience sampling alive is "I did not pick them for any particular reason, so the sample is basically random." Not choosing on purpose is not the same as choosing by chance. Randomness means a mechanism that hands every member of the population a known probability, and standing where people happen to be gives everyone else a probability of zero while the people who are there share whatever put them there.

A convenience sample is not a voluntary response sample, where the subjects put themselves in by answering. Here the researcher does the reaching. Both are nonrandom, and topic 1.12 Potential Problems with Sampling is where the course puts the damage they do.

One thing convenience recruiting does not wreck is an experiment. Randomly assigning treatments to volunteers who were easy to find still supports a cause-and-effect claim about people like those volunteers. What it cannot do is generalize to a wider population, because that right comes from random selection.