Before the method: the idea behind it
Learning about a whole from a part
Every opinion poll, medical trial and quality check faces the same problem: you want to know something about a large group, but you can only examine a few of them. This topic is about when that works and when it fails. It rests on two ideas that pull against each other: a sample must be chosen fairly, and even a perfectly fair sample will not match the population exactly.
Part one
Population, sample, parameter, statistic
Four words, and questions expect you to use them precisely.
The population is the entire group you want to know about. The sample is the part of it you actually examine. A number describing the population is a parameter, and it is usually unknown: that is the whole reason for sampling. The matching number computed from your sample is a statistic, and it is known but imperfect.
So if 63% of a sample of 500 voters support a policy, 63% is the statistic. The true population percentage is the parameter, and it is almost certainly not exactly 63%. The rest of this topic is about how far off it might be.
| Term | Meaning | Symbol for a proportion |
|---|---|---|
| Population | the whole group of interest | size N |
| Sample | the part examined | size n |
| Parameter | a number describing the population, usually unknown | p |
| Statistic | a number computed from the sample | p (read “p hat”) |
Part two
What makes a sample fair
A random sample is one where every member of the population has an equally likely chance of being selected.
That definition is doing real work. It is not enough for a sample to feel arbitrary or unplanned. Surveying the first 20 people who walk past feels random, but people out at that spot at that hour are not a fair cross-section of anyone. Methods that guarantee equal chances are called probability sampling methods.
Simple random
Every member of the population has an equal chance of selection.Number all 421 houses and use a random number generator to pick 100.
Systematic
Order the population by something unrelated to the question, then take every kth member.List employees alphabetically by surname and survey every 10th.
Stratified
Split into subgroups, then sample from each in proportion to its size.Sample 10% of each year group, so all years are represented fairly.
Cluster
Split into groups, then randomly select whole groups.Randomly choose 6 schools from a region and survey everyone in them.
Methods without that guarantee are non-probability methods, and they are where bias creeps in: convenience sampling takes whoever is easiest to reach, quota sampling stops once a target number is hit, and volunteer sampling lets people opt in.
Bias means the sample systematically misrepresents the population. It is not the same as random error: taking a bigger biased sample does not fix it, it just makes you more confident in a wrong answer.
One thing worth knowing before an exam: in these questions the name of the method or the bias earns little. What earns marks is describing the actual random process, or explaining precisely where the bias comes from and what effect it has.
Part three
Even fair samples disagree
Suppose you could see the whole population. Here 40% of it has the characteristic you care about, so p = 0.4.
Take a genuinely random sample and compute its proportion. Take another. They will not match, and neither will match 0.4 exactly. That is sampling variability, and it is not a mistake: it is unavoidable, and it is what the rest of this topic learns to measure.
The population: 200 people, 80 of them (teal) have the characteristic, so p = 0.4. Red outlines show who was sampled.
Take a few samples and watch how much the values scatter. Then increase the sample size and take more.
Two patterns should appear. The dots cluster around the true value of 0.4 rather than drifting off somewhere else, which is what makes random sampling trustworthy. And as you increase n, the scatter narrows: bigger samples give more reliable estimates.
Both observations turn into precise formulas in the next lesson. For now the important idea is that a single sample proportion is one draw from a whole distribution of possible values, and understanding that distribution is what lets you say how much to trust it.
Where this shows up in real life
Why polls disagree with each other
Two polls run the same week can differ by several points without either being wrong. Some of that gap is pure sampling variability, exactly what the dot plot shows. Reporting that treats every movement as a real shift in opinion is misreading noise as signal.
The 1936 polling disaster
A magazine mailed 10 million ballots and predicted a landslide for the wrong candidate, while a pollster using 50,000 carefully chosen people got it right. The huge sample came from car and telephone owners, who were wealthier than average. Size cannot rescue a biased sample.
Before you start
What to reach for
| The question asks… | Do this |
|---|---|
| identify the population, sample, parameter or statistic | Name each precisely, and say which is known |
| describe a procedure to ensure randomness | Describe the actual process, not just the method’s name |
| identify sources of bias | Say where it comes from and what effect it has on the results |
| comment on variability between samples | Expect scatter around p, narrowing as n grows |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide to running simulations.