Everyone or some?

Suppose you want to know what share of students in a school of 1 200 walk to school. You could ask everyone. Or you could ask 100 of them and use the answer to estimate the rest.

The words:

• Population: the whole group you want to learn about (all 1 200 students).
• Sample: the part of the population you actually measure (the 100 you asked).
• Parameter: a number that describes the population, such as the true share who walk. Usually unknown. Written p.
• Statistic: a number computed from the sample, such as the share of your 100 who walk. Written p̂ ("p-hat").

The whole idea of statistics fits in one sentence: we use a statistic to estimate a parameter we cannot afford to measure.

Sometimes the population can be measured completely. That is a census. A country's population count is an attempt at one. But measuring everyone is usually too slow, too costly, or impossible (you cannot test every light bulb until it burns out, or you would have none left to sell).

A sample is only an estimate, and different samples give different estimates. Imagine a town in which exactly 60 % of people like ice cream, and you ask 20 people at random. You might get 55 %. Next time 70 %. Next time 60 %. None of them is the truth, and none is "wrong"; this wobble from sample to sample is called sampling variability. The numbers in this example are invented for teaching. They are not real measurements.

Two things control how good an estimate is:

• How the sample was chosen. If the method leans toward certain people, the estimate leans with it. That is bias, and the next world is about it.
• How big the sample is. A larger random sample wobbles less. Lesson 3 of world 6 puts a number on that.

The best everyday method is a simple random sample: every member of the population has the same chance of being chosen, like drawing names from a hat. A random sample is not guaranteed to be perfect, but its mistakes are the kind we can measure.

A real example: the 344 penguins in the Palmer data set. Are they "all the penguins"? No. They are a sample from three kinds of penguin (Adelie, Chinstrap and Gentoo), living on islands near one research station, measured during three years. The researchers could not measure every penguin, and the data set does not claim to be a random sample of all penguins on Earth.

So when we find a pattern in these birds, we should say what it describes: these penguins. Whether it carries over to penguins elsewhere depends on how much the colonies near Palmer Station look like the rest, and that is a question about the sampling, not about the arithmetic.