Populations, Samples and Bias: Quick Revision

A brief note on Populations, Samples and Bias. Read it before a test. Try each question at the end before you open its answer.

The words

• Population: everyone or everything you want to know about.
• Sample: the part you really look at. A census looks at the whole population.
• Parameter: a number for the population, such as $\mu$ or $p$. Fixed, usually unknown.
• Statistic: a number from a sample, such as $\bar x$ or $\hat p$. Known, and it varies from sample to sample.
• Sampling variability: fair samples differ by luck. A sample four times as large halves the usual error.

The kinds of bias

• Convenience: asking whoever is easy to reach.
• Voluntary response: people choose themselves, and strong views answer most.
• Undercoverage: some groups can never be chosen.
• Survivorship: only the ones that came through a filter are studied.
• Non-response: chosen people do not answer, and they differ from those who do.
• Response bias: leading, loaded or double questions push the answers.

Two formulas

• Truth with two groups: $\dfrac{f a + (100 - f) b}{100}$, when $f\%$ are in the first group.
• Share of a group in the replies: $\dfrac{p\,r_y}{p\,r_y + (100 - p)\,r_n} \times 100\%$.

A worked example

In a made-up town, 20% of the adults run, and 75% of the runners want a new track. Of the others, 25% want it. A survey asks runners only.

1. The truth. $\dfrac{20 \times 75 + 80 \times 25}{100} = \dfrac{1500 + 2000}{100} = 35\%$.
2. The survey. About 75%.
3. The bias. $75 - 35 = 40$ percentage points.

Practice with new numbers

A program to try

The program works out the truth and the bias of the worked example.

Mistakes to avoid

• Trusting size. Size cures luck, never bias.
• Calling luck "bias". Bias is a slant in the method, not one unlucky result.
• Forgetting who did not answer. Ask who is missing, not only who is there.

Check yourself

1. Is "the mean age of all the members of a club" a parameter or a statistic?

2. A random sample of 50 phones from a batch of 4000 has 3 faulty phones. Estimate the number of faulty phones in the batch.

3. Name the bias: a survey of commuters phones only landline numbers.

4. Supporters make up 50% of a town. They reply 30% of the time, and the others 10% of the time. What share of the replies support?

Sources

• OpenStax, Introductory Statistics, Chapter 1, sections on data, sampling and variation in data and sampling.