A statistical question is often about a big group: all the students in a school, all the voters in a town. Asking everyone is usually too slow or too costly, so we ask a smaller group and use it to learn about the whole.
Key idea
The whole group is the population; the part you study is a sample. In a random sample, every member of the population has the same chance to be chosen. Random samples tend to represent the population well. Samples chosen for convenience can be biased.
Why it works
If some members are more likely to be picked, the sample leans toward them. Ask only your friends how much they like your favorite band, and the answers lean high. Ask people at a gym how often they exercise, and the answers lean high too. Random choice gives no group an edge, so the sample's mix is likely to be close to the population's mix.
| Plan | Random? | Why |
|---|---|---|
| Number all 800 students and let a computer pick 60 numbers | Yes | Every student has the same chance |
| Ask 60 friends | No | Friends are often alike |
| Ask the first 60 people to answer an online poll | No | People choose themselves; strong opinions answer more |
| Ask everyone in one classroom | No | One class may not be like the whole school |
Worked examples
Worked example: Choosing a plan
A principal wants to know how 7th graders get to school. Which plan gives a random sample: (a) ask the students in the bike club, or (b) draw 50 names from a box holding every 7th grader's name?
- In plan (b), every 7th grader has the same chance to be drawn.
- Plan (a) leans toward students who bike.
Answer: Plan (b)
Worked example: Big but biased
A website poll gets 5,000 answers, and 85% say they like video games. Does this show that 85% of all people like video games?
- The people who answered chose to visit that site and to reply.
- They are likely to be more interested in video games than other people, so the sample is biased.
- A large biased sample is still biased.
Answer: No
Worked example: In a factory
An inspector checks light bulbs for defects. Should she test the first 50 bulbs made in the morning, or 50 bulbs picked at random times through the day?
- The machine could change during the day, so the morning bulbs may not be typical.
- Bulbs picked at random times give every bulb the same chance.
Answer: Random times through the day
Common mistake
Trusting a sample because it is large. Size helps only when the sample is random. A huge sample of volunteers or friends can be far off. Ask first: did every member of the population have the same chance to be chosen?