Using Data
You almost never get to survey everyone. If a school wants to know how its 800 students feel about a later start time, nobody asks all 800 — they ask a smaller group and use the answers to estimate. The entire group you want to know about is the population, and the smaller group you actually collect data from is the sample.
Two skills make this work: judging whether a sample is fair, and scaling the sample's results up to the population with a proportion. Both show up constantly on tests, and both take less than a minute once you see the pattern.
Population vs. sample
The population is everyone the question is about — all 800 students, all voters in a city, every phone off the assembly line. The sample is the part of that group you actually surveyed or measured.
A quick check: the population matches the question being asked, not the people you happened to talk to. If the question is about the whole school, the population is the whole school even if you only surveyed 30 students.
What makes a sample fair
A sample is fair only when it is random — every member of the population has an equal chance of being picked. If the way you gathered the sample favors certain people, the sample is biased and its results cannot be trusted.
Bias usually hides in the location or timing of the survey. Asking 30 students at 7 a.m. band practice about a later start time is biased: only students who already show up early can be chosen, and they may feel differently than everyone else. The problem is not the sample's size — it is who could get picked.
Predicting with a proportion
If the sample is random, the fraction you see in the sample should roughly match the fraction in the population. Set the two fractions equal: , then solve for .
For example, if of sampled students ride the bus, then predicts of the school's students ride the bus. The prediction is an estimate, not a guarantee — but with a random sample, it is a good one.
Worked examples
Example 1: identify the population and the sample
A factory produces 5,000 light bulbs a day. A worker tests 100 of them for defects. Name the population and the sample.
Answer: Population: all bulbs; sample: the tested bulbs
Example 2: spot the bias
To learn how a town of 20,000 people feels about building a new gym, a surveyor asks 50 people leaving a fitness store. Is the sample random?
Answer: No — the sample is biased toward people who already care about fitness
Example 3: predict with a proportion
In a random sample, of students ride the bus. Predict how many of the school's students ride the bus.
Answer: About students
Try one yourself
Common questions
Why not just survey the whole population?
Usually it costs too much, takes too long, or is impossible — you cannot crash-test every car. A well-chosen random sample gives a reliable estimate for a fraction of the effort.
Does a bigger sample fix a biased sample?
No. If the selection method leaves some people out, adding more people chosen the same way just repeats the mistake at a larger scale. Fix how the sample is chosen first; then a bigger sample helps.
Is a prediction from a sample exact?
No — it is an estimate. A random sample of 40 students predicts about 180 bus riders out of 600, but the true count could be somewhat higher or lower. Random sampling makes the estimate trustworthy, not perfect.
What does 'random' actually mean here?
Every member of the population has an equal chance of being selected. Names drawn from a hat or numbers picked by a computer are random; whoever happens to walk by is not.
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