Random Sampling & Representative Samples
You can't ask every student in a school, every voter in a state, or test every part off an assembly line. So statisticians ask a sample — a smaller group — and use its answers to describe the whole population. That shortcut only works if the sample actually looks like the population.
A sample is representative when every member of the population has an equal chance of being chosen. The reliable way to get that is random selection — names from a hat, every 10th customer, a random draw from the full roster. A random sample lets you generalize to the whole population; a biased one doesn't.
Representative vs. biased samples
Ask one question about any sample: did everyone in the population have an equal chance of being picked? If yes, the sample is representative. If some group was more likely to be included — or left out entirely — the sample is biased.
Surveying only the soccer team about the new schedule leaves out everyone who isn't on the team. Surveying the first students to arrive favors early risers. Surveying one classroom leaves out every other class. Each of those samples overrepresents one kind of student, so what it says may not match the school.
Sample size doesn't fix bias. Asking soccer players is still a soccer-team opinion. A random sample of from the full roster beats a biased sample of any size.
Predicting a population from a sample
Once you have a random sample, you can scale its results up to the population, because the fraction in a representative sample should match the fraction in the population.
If of sampled students ride bikes, the sample fraction is . Set it equal to the population fraction and solve: . It's a proportion, and everything you know about solving proportions applies.
Remember these are estimates. A prediction of riders means about — another random sample would give a slightly different number, and that's normal.
Worked examples
Example 1: pick the representative sample
A school wants students' favorite sport. Which sample is representative: asking only the basketball team, or drawing names from a hat containing every student's name?
Answer: Drawing names from the hat is the representative sample
Example 2: predict from a sample
In a random sample of students, ride bikes to school. The school has students. Predict how many students ride bikes.
Answer: About students ride bikes
Example 3: a messier proportion
In a random sample of students, pack their lunch. The school has students. Predict how many students pack lunch.
Answer: About students pack their lunch
Try one yourself
Common questions
Why does random selection make a sample representative?
Random selection gives every member an equal chance, so no group is systematically favored or left out. On average, the sample ends up with the same mix as the population — same fraction of athletes, early risers, and everyone else.
Is a bigger sample always better?
Bigger helps only if the sample is random. A larger random sample gives steadier estimates. But a huge biased sample just repeats the same bias louder — 500 soccer players still only tell you what soccer players think.
Is surveying every 10th customer really random?
It's called a systematic sample, and at this level it counts as representative: every customer has an equal chance, since who arrives 10th, 20th, 30th is effectively chance. The samples to reject are the ones favoring a group — volunteers, friends, one class.
Will the prediction match the real count exactly?
Almost never, and it isn't supposed to. A prediction of bike riders means about . Different random samples give slightly different predictions, all clustered near the truth — that cluster is what makes sampling useful.
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