Populations, Samples, and Statistical Studies

Every statistical study starts with a question about some population: all registered voters, every light bulb a factory makes, all the fish in a lake. Measuring the whole population is usually impossible, too slow, or destructive, so researchers measure a sample instead, a subset actually selected and observed. Any numerical fact about the population, like the true proportion of all voters who support a candidate, is a parameter. It is usually unknown and often unknowable. The matching number calculated from the sample, like the proportion of voters in your poll who support that candidate, is a statistic. You calculate statistics because you want to estimate parameters.

The key move is random selection. If a sample is chosen randomly, it tends to look like the population on average, so its statistic becomes a reasonable stand-in for the parameter. This is why a well-chosen sample of 1,000 people can represent 300 million: randomness, not size relative to the population, is what makes the sample trustworthy. What actually controls how close the statistic lands to the parameter is the absolute sample size. Bigger samples give statistics that vary less from one sample to the next, a property called sampling variability, regardless of how small a slice of the population they are.

This idea sets up everything else in the unit. Once you can tell populations from samples and parameters from statistics, you are ready to ask how samples should be collected, through random sampling methods, and later how much a statistic might bounce around just by chance, which leads into sampling distributions.