Populations, Samples and Variables

Main idea

Statistics is about learning something true of a whole group from a small part of it. The number you want, for everyone, is a parameter, and you almost never get to see it; the number you can actually compute from your sample is a statistic, your best estimate of it.

{\htmlData{sym=0}{\bar x}} \approx {\htmlData{sym=1}{\mu}}, \qquad {\htmlData{sym=2}{\hat p}} \approx {\htmlData{sym=3}{p}}

What the symbols mean

$\bar x$
Sample mean
A statistic: the average of the data you collected.
$\mu$
Population mean
A parameter: the average over everyone, usually unknown.
$\hat p$
Sample proportion
The fraction of the sample with some property.
$p$
Population proportion
The fraction of the whole population with it.

Explanation

Statistics is simply the craft of learning about a big group (all voters, every phone a factory makes) by carefully measuring a small part of it. The formulas here are just two pairs of names: $\bar x \approx \mu$ (the average in your sample stands in for the average of everyone) and $\hat p \approx p$ (the percentage in your sample stands in for the percentage of everyone). The rule of thumb for the notation: Greek letters and plain $p$ are the true values for the whole population, and letters with a bar or a hat are what you computed from your sample. That's it for symbols in this lesson.

Vocabulary:

Population: everyone or everything you want to know about.
Sample: the part you actually measure.
Variable: a characteristic that varies between individuals, like height or blood type.
Data: the observed values of a variable.

Why not just measure the whole population? Usually it's impossible or far too expensive: you can't interview every voter, and you can't crash-test every car you sell. So we measure a sample and reason carefully about how far off it might be.

Kinds of variables:

Qualitative (categorical): labels, like blood type or eye color.
Quantitative discrete: counts, like number of siblings.
Quantitative continuous: measurements on a continuum, like time or weight.

Sampling variability. Different samples give different statistics. Poll 1,000 people today and 1,000 different people tomorrow, and you'll get two slightly different percentages, even though the population hasn't changed. A good sample's statistics cluster around the true parameter, and larger samples cluster more tightly. Quantifying that variability is what most of statistics is about, and we'll spend much of this course on it.

A common mistake. It's tempting to treat a statistic as if it were the parameter: "52% of those polled support it, so 52% of voters do." The 52% is one sample's estimate. The true value is probably close, but another sample would have given a somewhat different number, and the rest of this course is about saying how close.

See it

In this population 30% would say yes, the parameter p, which we don't get to see. Each sample of n people gives its own p̂, and the curve shows how those p̂ values spread from sample to sample. With n = 10, 95% of samples land within about ±28 points of the truth, which is nearly useless. By n = 1000 it's within about ±3 points: 100 times the sample buys 10 times the precision.

Worked examples

Example 1

A poll of 1,000 voters finds 52% support a measure. Identify the population, sample, parameter and statistic.

1
The population is everyone we want to know about, all voters. The sample is the 1,000 we actually asked.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

In a poll of 1,000 voters, 52% support a measure. Is 52% a parameter or a statistic?

Problem 2+5 XP

"Number of siblings" is which kind of variable?

Problem 3+5 XP

"Time to run a mile" is which kind of variable?

Problem 4+5 XP

"Blood type" is which kind of variable?

Problem 5+5 XP

In a sample of 200 students, 46 own a bike. Find the sample proportion.

Problem 6+5 XP

A quality inspector checks 80 phones from a day's production of 12,000 and finds 6 defective. Estimate how many phones in the day's production are defective.

Problem 7+5 XP

The typical sample-to-sample spread of a sample proportion is its standard error, $\sqrt{\frac{p(1 - p)}{n}}$ . Find it for $p = 0.3$ and $n = 100$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

How large a random sample guarantees a standard error of at most 0.01 for $\hat p$ , whatever the true $p$ is?

Where it's used

Election polls, clinical trials, quality inspections and A/B tests on websites all measure a sample to make decisions about a population.

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