From sample to population

Gallup’s 2015 finding that “a slight majority of Americans, 54%, say it is extremely important that parents get their children vaccinated” makes a claim about the population of 258 million adults. But Gallup doesn’t know the truth about the entire population. The poll contacted 1015 people and found that 54% of them said it is extremely important for parents to vaccinate their children. Because the sample of 1015 people was chosen at random, it’s reasonable to think that they represent the entire population pretty well. So Gallup turns the fact that 54% of the sample feel childhood vaccinations are extremely important into an estimate that about 54% of all American adults feel childhood vaccinations are extremely important. That’s a basic move in statistics: use a fact about a sample to estimate the truth about the whole population. To think about such moves, we must be clear whether a number describes a sample or a population. Here is the vocabulary we use.

Parameters and statistics

A parameter is a number that describes the population. A parameter is a fixed number, but in practice we don’t know the actual value of this number.

A statistic is a number that describes a sample. The value of a statistic is known when we have taken a sample, but it can change from sample to sample. We often use a statistic to estimate an unknown parameter.

So parameter is to population as statistic is to sample. Want to estimate an unknown population parameter? Choose a sample from the population and use a sample statistic as your estimate. That’s what Gallup did.

EXAMPLE 1 Should children be vaccinated?

The proportion of all American adults who feel childhood vaccinations are extremely important is a parameter describing the population of 258 million adults. Call it p, for “proportion.” Alas, we do not know the numerical value of p. To estimate p, Gallup took a sample of 1015 adults. The proportion of the sample who favor such an amendment is a statistic. Call it , read as “p-hat.” It happens that 548 of this sample of size 1015 said that they feel childhood vaccines are extremely important, so for this sample,

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Because all adults had the same chance to be among the chosen 1015, it seems reasonable to use the statistic as an estimate of the unknown parameter p. It’s a fact that 54% of the sample feel childhood vaccines are extremely important—we know because we asked them. We don’t know what percentage of all American adults feel this way, but we estimate that about 54% do.