Convenience Sample vs Simple Random Sample

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.

A simple random sample of size $n$, usually shortened to SRS, comes from a method under which every possible sample of $n$ individuals is equally likely to be the one drawn. The condition is about whole samples, not about individuals. Equal chance for each individual follows from it but does not imply it, and that gap is where most naming mistakes live.

With 5 students and $n = 2$ there are 10 possible samples, each of probability $1/10$, and each student sits in 4 of the 10, so every individual has chance $4/10 = 2/5$. Now take a different plan for four students A, B, C and D: flip one fair coin, take A and B on heads, take C and D on tails. Every student still has chance $1/2$, and the sample size is still 2, but only 2 of the 6 possible pairs can ever come up. The pair A and C has probability 0. The plan uses chance, and it is not an SRS.

So distrust the sentence "everyone had an equal chance of being picked, so it was a simple random sample." Equal individual chance is necessary and not sufficient. A cluster sample gives every individual the same chance while making most subsets impossible, which is why the SRS condition is written in terms of samples.

Carrying one out needs a list. Number a roster of 500 students 1 to 500, generate random integers in that range, ignore repeats, and take the first 30 distinct labels. If the roster leaves people out, the result is an SRS of the sampling frame and still biased for the population, because no amount of randomness inside a short list reaches anyone who is not on it.

Random selection is topic 1.11 Random Sampling in Unit 1. It is what supports generalizing to the population; random assignment is the separate move that supports a cause-and-effect claim.