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Populations and samples

Statistics and data · level 5, lesson 4 of 6

A question to start

A doctor wants to know your blood type. Does she take all of your blood to test it? Of course not. She takes a few drops, and she is completely confident that the rest of your blood is the same.

A news report says "38% of Canadians plan to travel this summer". Canada has about 40 million people. Nobody asked all of them. Perhaps one or two thousand people were asked. How can so few people tell us about so many, and when should we believe it?

The idea

Population and sample

The population is the whole group you want to know about. It might be people (every student in your school, every adult in Canada), but it can be anything: every light bulb made in a factory this week, every lake in Ontario, every rock in a crater.

A sample is the part of the population that you actually look at.

population: all 60 sample: 8

The whole point of a sample is to learn about the population without looking at all of it. A good sample is like a small copy of the population: if 30% of the population likes a thing, roughly 30% of a good sample will too.

Census or sample?

A census collects data from every member of the population. A sample survey collects data from only a sample. Why not always do a census?

  • Cost and time. Asking 40 million people is enormously expensive and slow. By the time you finish, the answer may have changed.
  • Destruction. To find out how long a light bulb lasts, you must burn it until it fails. To test a car's safety, you crash it. A census of every bulb would leave nothing to sell.
  • Impossibility. You cannot catch every fish in a lake, or find every person with a rare illness.
  • Accuracy. Surprisingly, a carefully run sample can be more accurate than a rushed census, because a smaller team can check every answer carefully.

A census makes sense when the population is small (your class of 25) or when you truly need every member (a country counting its people to decide how many seats each region gets in Parliament).

Canada's census

Statistics Canada, the country's national statistics office, runs a Census of Population every five years: 2016, 2021, 2026. Every household must answer a short questionnaire (how many people live there, their ages, and a few more questions). That part is a true census.

But even the census uses sampling. A longer, more detailed questionnaire (about work, education, travel to work, and much more) goes to only about one household in four. The answers from that sample are used to describe the whole country. In 2011 the long questionnaire was made voluntary, and in 2016 the required long form came back; you will see in lesson 6 why a voluntary survey worries statisticians.

Parameters and statistics

Two words keep population numbers and sample numbers apart:

  • A parameter is a number that describes the population: the true percent of all Canadians who plan to travel, the true mean height of all 13-year-olds in your province. It is usually unknown.
  • A statistic is a number calculated from the sample: the percent of the 1500 people asked who plan to travel. It is known, and we use it to estimate the parameter.

A memory trick: parameter goes with population, statistic goes with sample.

From a part to the whole

If a sample is a fair small copy, you can scale up. Suppose 50 students out of a school of 800 are chosen fairly, and 18 of them walk to school. The sample proportion is \(18 \div 50 = 0.36\), or 36%. So we estimate that about 36% of the whole school walks: \(0.36 \times 800 = 288\) students.

This is just proportional reasoning. It is only as good as the sample: if the 50 students were all from the street next to the school, the estimate would be far too high.

Samples vary

Here is the catch. Different samples from the same population give different statistics, just by luck of which members were chosen. This is called sampling variability.

Imagine a tiny population: ten friends who read 2, 3, 4, 4, 5, 6, 7, 8, 9 and 12 books last summer. The population mean (a parameter) is \(60 \div 10 = 6\) books. Now take some samples of 4 friends:

Sample Sample mean
2, 4, 7, 12 6.25
3, 4, 5, 6 4.5
7, 8, 9, 12 9
4, 5, 8, 9 6.5

None of them is "wrong". They are four honest estimates, scattered around the true value 6.

What happens with bigger samples? Here are all the possible samples of each size from these ten friends:

Sample size Number of possible samples Smallest sample mean Largest sample mean Share within 1 book of 6
2 45 2.5 10.5 about 44%
4 210 3.25 9 about 63%
8 45 4.875 6.875 about 98%

The bigger the sample, the closer the sample means huddle around the truth. That is the reason a poll of 1500 well-chosen people can be trusted to within a few percentage points. You will measure exactly how close in level 8.

A sample from Mars

Scientists face the population-and-sample problem everywhere. NASA's Perseverance rover, exploring Jezero crater on Mars since 2021, cannot bring the crater home. It drills small rock cores, each a little bigger than a piece of chalk, seals them in tubes, and scientists hope that a later mission will bring some of them back to Earth. Each core is a sample; the rocks of the crater are the population. The science team chooses where to drill with great care, because a few dozen cores must stand for a whole landscape.

Worked examples

1. Naming the parts. A juice company wants to know the mean amount of juice in all the bottles filled today. A worker measures 40 bottles and finds a mean of 998 mL.

  • Population: all bottles filled today.
  • Sample: the 40 measured bottles.
  • Parameter: the mean amount in all of today's bottles (unknown).
  • Statistic: 998 mL, the mean of the 40 bottles.

2. Census or sample?

  • The favourite lunch of the 24 students in your class: a census, it is easy to ask everyone.
  • How long batteries of a new kind last: a sample, because testing uses up the battery.
  • The opinions of all voters in a province: a sample, a census would cost far too much.

3. Scaling up. A random sample of 400 adults in a town of 9000 adults finds that 120 own a bicycle. Estimate the number of bicycle owners in the town.

Sample proportion: \(120 \div 400 = 0.3\). Estimate: \(0.3 \times 9000 = 2700\) adults. (Because of sampling variability, we would say "about 2700", not "exactly 2700".)

4. Counting fish you cannot see. Biologists want to know how many fish live in a pond. They catch 50 fish, put a small harmless tag on each, and let them go. A week later, after the fish have mixed, they catch 40 fish and find that 8 are tagged.

In the second catch, 8 out of 40, which is \(\frac{1}{5}\), are tagged. If the catch is a fair sample, then about \(\frac{1}{5}\) of all the fish in the pond are tagged. There are 50 tagged fish, so the pond has about \(50 \times 5 = 250\) fish. This method is called capture-recapture, and it is really used to estimate animal populations.

Try it

  1. A school wants to know how many hours of sleep its 600 students get. It asks 60 students and finds a mean of 8.1 hours. Name the population, the sample, the parameter and the statistic.
  2. Would you use a census or a sample? (a) The ages of everyone in your family. (b) Whether a new batch of fireworks works. (c) The favourite sport of all teenagers in Canada.
  3. In a fair sample of 250 households in a city of 60 000 households, 35 have a dog. Estimate the number of households with a dog.
  4. Biologists tag 60 frogs and release them. Later they catch 50 frogs, and 12 are tagged. Estimate the number of frogs.
  5. Two classmates each take a fair sample of 30 students from your school and find mean heights of 158 cm and 161 cm. Has one of them made a mistake? Explain.
  6. Using the ten readers from the lesson, find the mean of the sample 3, 5, 8, 12. How far is it from the population mean?
  7. Why can a sample sometimes be more accurate than a census?
Answers
  1. Population: all 600 students. Sample: the 60 who were asked. Parameter: the mean sleep of all 600 (unknown). Statistic: 8.1 hours.
  2. (a) Census: the population is tiny. (b) Sample: testing a firework uses it up. (c) Sample: there are millions of teenagers.
  3. \(35 \div 250 = 0.14\), and \(0.14 \times 60\,000 = 8400\) households, approximately.
  4. 12 out of 50 caught are tagged, so the tagged frogs are about \(\frac{12}{50}\) of all frogs. If \(N\) is the number of frogs, \(\frac{60}{N} = \frac{12}{50}\), so \(N = \frac{60 \times 50}{12} = 250\) frogs.
  5. Not necessarily. Different samples give different statistics just by chance (sampling variability). A difference of 3 cm between two samples of 30 is quite ordinary.
  6. \((3 + 5 + 8 + 12) \div 4 = 28 \div 4 = 7\) books, which is 1 book above the population mean of 6.
  7. A census is so big that it is hard to check every answer, and mistakes or missing answers creep in. A smaller sample can be collected and checked very carefully.

Common mistake

Treating a statistic as the exact truth. A sample's 36% is an estimate of the population's percent. Another fair sample might give 33% or 39%. Say "about", and remember that the estimate is only good if the sample was chosen fairly, which is the next lesson.

Thinking a sample must be a big fraction of the population. What matters most is how the sample was chosen and how many members it has, not what fraction of the population it is. A fair sample of 1500 works about as well for a city as for a whole country.

Where it leads