Fair Ways to Choose a Sample: Quick Revision

A brief note on Fair Ways to Choose a Sample. Read it before a test. Try each question at the end before you open its answer.

The four methods

• Simple random: every group of $n$ is equally likely. Each member has the chance $\dfrac{n}{N}$. In Python: random.sample(frame, n).
• Systematic: step $k = \dfrac{N}{n}$, a random start, then every $k$th member. Beware of a pattern in the list.
• Stratified: a simple random sample inside each stratum. Proportional allocation: $n \times \dfrac{N_i}{N}$ places for stratum $i$.
• Cluster: whole groups chosen at random, everyone in them asked. Cheap, but less precise when clusters are alike.

The sampling frame

• The frame is the list you choose from. People missing from it can never be chosen.

The margin of luck

• Quick rule: margin $\approx \dfrac{100}{\sqrt n}$ percentage points, for about 95 polls in 100.
• Sample size: $n = \left(\dfrac{100}{m}\right)^2$ for a margin $m$.
• Halving the margin needs four times the sample.
• The margin covers luck only, never bias.

A worked example

A college has 600 science students and 400 arts students. It wants a stratified sample of 50.

1. Science places. $50 \times \dfrac{600}{1000} = 30$.
2. Arts places. $50 \times \dfrac{400}{1000} = 20$.
3. The margin for the whole sample. About $\dfrac{100}{\sqrt{50}} \approx 14.1$ points, so a sample of 50 gives only a rough answer.

Practice with new numbers

A program to try

The program shares the places of the worked example and finds the margin.

Mistakes to avoid

• Strata or clusters? Strata: some from every group. Clusters: all from some groups.
• Repeats. random.choices can repeat a member. Use random.sample.
• Unweighted strata. Equal samples from unequal strata must be weighted.

Check yourself

1. A list of 750 names needs a systematic sample of 25. What is the step?

2. Find the margin of luck of a fair poll of 400.

3. A firm picks 5 of its 60 offices at random and asks every worker in them. Which method is this?

4. How many people give a margin of 10 percentage points?

Sources

• OpenStax, Introductory Statistics, Chapter 1, section on data, sampling and variation, and Chapter 8, on confidence intervals for a proportion.