Populations vs Samples

Suppose you want the average commute time of every worker in a city. There are millions of them, they change jobs, and nobody has time to ask them all. Measuring the entire group is almost never possible — too expensive, too slow, sometimes physically impossible (you cannot test every light bulb by burning it out).

So statistics does something clever: it measures a small slice, then reasons carefully back to the whole. Getting that reasoning right starts with naming the two groups precisely.

Once you have a population and a sample, you will compute the same kinds of numbers — an average, a spread — for each. They get different names and different symbols so nobody confuses the true value with the estimate.

• A number that describes the population is a parameter. It is usually unknown and fixed. The population mean is \( \mu \) ("mu") and the population standard deviation is \( \sigma \) ("sigma").
• A number computed from your sample is a statistic. You can actually calculate it, and it changes from sample to sample. The sample mean is \( \bar{x} \) ("x-bar") and the sample standard deviation is \( s \).

A handy memory hook: parameter goes with population; statistic goes with sample. Greek letters describe the unknown truth; ordinary letters describe what you measured.

The simulation below draws samples from a population whose true shape is fixed. The top strip is the whole population. Each time you draw, a handful of values is pulled out and its average recorded in the bottom panel.

For this lesson, focus on one idea: a single sample's average lands near the population mean but rarely exactly on it, and different samples give slightly different averages. That wobble — the fact that a statistic like \( \bar{x} \) changes from sample to sample — is exactly what the next lessons on sampling distributions are about.

A sample fails when it is systematically unlike the population — that is sampling bias. Bias is not the same as bad luck; it is a lean in one direction that repeats no matter how much data you collect.

The most common culprit is a convenience sample — grabbing whoever is easy to reach. Ask about commute times only in the office parking lot at 9am and you miss everyone who works from home, takes the train, or starts at noon. Poll about a product only among people who already follow your brand and you hear mostly fans. The sample is easy to get and quietly skewed.

The cure is to remove your choices from the selection. In a simple random sample, every member of the population has an equal chance of being picked, and picks do not depend on who you find convenient. Randomness has no agenda, so on average the sample mirrors the population — fast and slow commuters show up in roughly their true proportions.

Random selection does more than remove bias. Because the picks are random, the sample statistic behaves in a predictable, mathematical way from sample to sample. That predictable behavior — the sampling distribution — is what lets statistics attach honest margins of error to an estimate, and it is exactly where the next lessons go.