Math Core

Suppose you want to know how many hours a week the students at your school spend on homework. Asking every single student would take forever. Instead, you can ask a smaller group and use their answers to learn about everyone. The trick is choosing that smaller group fairly.

Populations and samples

Definition

Population and sample

The population is the entire group you want to learn about. A sample is the part of the population you actually collect data from.

If you want to know the favorite lunch of the $600$ students at your school, the population is all $600$ students. If you ask $50$ of them, those $50$ students are your sample.

Statisticians use samples all the time. A company can't test every battery it makes (testing uses up the battery!), and a news poll can't call every voter in the country. A good sample lets you make a reasonable guess about the population without collecting data from everyone.

Representative samples

A sample is only useful if it looks like the population. A representative sample has about the same mix of people or things as the whole population.

Imagine you ask only the students in the chess club about their favorite after-school activity. You'll probably hear "chess" a lot, but that doesn't tell you much about the whole school. This sample is biased: it tends to favor certain answers.

Here are common ways a sample becomes biased:

Convenience samples. You ask whoever is easiest to reach, such as your friends or the people in one classroom.
Voluntary response samples. People choose whether to answer, such as an online poll. People with strong opinions are more likely to respond.
Sampling the wrong place. Asking people leaving a gym how often they exercise will overestimate how much the whole town exercises.
Leading questions. A question like "Don't you agree that recess should be longer?" pushes people toward one answer.

Random samples

Random sampling

In a random sample, every member of the population has an equal chance of being chosen. Random samples tend to be representative, so they give the most trustworthy information about a population.

Randomness keeps your own choices (and your own opinions) out of the picture. Here are a few ways to pick a random sample:

Write every name on an identical slip of paper, mix the slips in a bag, and draw without looking.
Give every member a number, then use a random number generator to pick numbers. Skip any repeats.
Take a random starting point in a list, then choose every $10$ th name after it (this is called a systematic sample).

Worked example: Is it random?

A principal wants to know whether students at her school of $900$ want a later start time. Decide whether each method gives a random sample.

She asks the first $40$ students who arrive at school in the morning.
She uses a computer to randomly choose $40$ student ID numbers from the list of all $900$ students.
She posts a survey on the school website, and $40$ students fill it out.

Solutions.

Not random. Students who arrive early might already like getting up early, so they may be less likely to want a later start.
Random. Every student has the same chance of being picked.
Not random. This is a voluntary response sample. Students who care a lot about the start time are more likely to answer.

Worked example: Spotting bias

A town wants to know whether residents support building a new dog park. A volunteer surveys $100$ people walking their dogs in the town square. Explain why the results might be biased, and suggest a better method.

Solution. People walking dogs are far more likely to want a dog park than the typical resident, so the survey will probably overestimate support. A better method is to randomly choose $100$ addresses from a list of all homes in town and survey one adult at each.

Keeping groups in proportion

Sometimes a population is made of groups, and you want each group to show up in your sample in the right amount.

Worked example: A proportional sample

A middle school has $240$ sixth graders, $200$ seventh graders and $160$ eighth graders. The student council wants a sample of $60$ students with each grade represented in proportion to its size. How many seventh graders should be in the sample?

Solution. The school has $240 + 200 + 160 = 600$ students. Seventh graders make up $\dfrac{200}{600} = \dfrac{1}{3}$ of the school. So one third of the sample should be seventh graders:

\dfrac{1}{3} \times 60 = 20 \text{ seventh graders}.

The council would then randomly choose $20$ seventh graders from the list of all seventh graders. (Similarly, it would choose $24$ sixth graders and $16$ eighth graders, for a total of $60$ .)

Common mistake

A bigger sample is not automatically a better one. An online poll with $5{,}000$ volunteers can be more misleading than a random sample of $100$ , because the volunteers aren't representative. First make the sample random; then a larger random sample gives more precise results.

Tip

To test a sampling method, ask yourself: "Is there any reason the people chosen might answer differently from everyone else?" If the answer is yes, the sample is probably biased.

Practice

Practice 1

A factory wants to check the quality of the $12{,}000$ bottles it filled today. A worker tests $150$ of them. What is the population?

Practice 2

A club with $80$ members wants to choose $10$ members for a survey. Which method gives a random sample?

Practice 3

To estimate how many books students at a school read each year, Jalen surveys students at the school library after school. Why is this sample likely biased?

Practice 4

Which survey question is least likely to push people toward a particular answer?

Practice 5

A school has $840$ students listed in alphabetical order. The office picks a random starting name among the first $10$ and then chooses every $10$ th student on the list. How many students will be in the sample?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A school has $300$ seventh graders and $200$ eighth graders. The principal wants a sample of $50$ students in which each grade is represented in proportion to its size. How many seventh graders should be in the sample?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Ms. Rivera numbers each of the $450$ students in her school from $1$ to $450$ . She uses a random number generator to choose $30$ numbers. The generator gives her the number $17$ twice. What should she do?

Practice 8

A city council wants to know whether residents support a new bike lane. Which plan will give the most trustworthy results?