11.1 Population, sample and informal inference — Learn

Prerequisites: reading a list of data, finding an average from Number and Statistics at GCSE, and using a fraction as a proportion.

By the end of this section you can: explain the terms population, sample and sampling frame; say why a sample is often used instead of a census; use a sample mean or proportion to make an informal inference about the population; and explain why two samples from the same population can give different answers.

1. The population and the sample

A population is the whole group you want to know about. The population does not have to be made of people. It could be every bus journey on a route, every light bulb in a factory, or every tree in a wood.

A sample is a part of the population that you actually measure. A census measures every member of the population, and a sample measures only some of it.

For the bus company problem, the population is every journey made by that 07:30 service. The sample is the 30 journeys the passenger group timed. The group wants to know a fact about the population, such as the mean journey time. The mean of the sample is the best estimate it has.

Watch out. A sample is not a small population. The 30 journeys are 30 real journeys, but they are chosen to stand in for the whole service. When you write a conclusion, say which group it describes. "The mean of the sample is 25 minutes" and "the mean of all journeys is 25 minutes" are different claims.

2. Why we use a sample

A census gives the exact answer, so why take a sample?

Time and cost. Measuring a whole population can take months.
Access. You may not be able to reach every member of the population.
Testing can destroy the item. If you test a match to its end, you cannot sell it afterwards.
A sample can be accurate enough. A well-chosen sample often gives a good estimate of the population.

3. The sampling frame

A sampling frame is the list of the population that you choose from. For the bus company, a frame could be every journey in the timetable for one month. For a school survey, it could be the school register.

The frame matters. If the frame leaves out part of the population, the sample cannot represent that part. A frame of landline numbers leaves out homes with no landline, so a telephone survey of those homes would miss them.

4. Informal inference

An informal inference is a conclusion about a population drawn from a sample. It is informal because no probability statement is attached to it yet. The tools from Chapter 13 add the probability later.

Two simple estimates are used again and again.

Estimate a population total. Multiply the sample mean by the number in the population.

\text{estimated total} = \text{sample mean} \times \text{population size}.

Estimate a population count. Multiply the population size by the sample proportion.

\text{estimated count} = \text{population size} \times \frac{\text{number in the sample with the feature}}{\text{sample size}}.

Example 1. A sample of 40 students spends a mean of $3.5$ hours per week on homework. The school has 1200 students. Estimate the total number of hours spent on homework each week.

\text{estimated total} = 3.5 \times 1200 = 4200 \text{ hours}.

Example 2. In a sample of 200 components, 12 are faulty. A batch contains 5000 components. Estimate the number of faulty components in the batch.

The sample proportion is

\frac{12}{200} = 0.06.

So the estimated count is

0.06 \times 5000 = 300 \text{ components}.

Example 3. In a sample of 80 shoppers, 52 prefer brand A. Estimate the proportion of all shoppers who prefer brand A.

\frac{52}{80} = 0.65.

About $65\%$ of shoppers prefer brand A. This is an estimate from one sample, not a measurement of every shopper.

5. Different samples give different answers

Suppose the passenger group timed a second week of journeys. The second sample of 30 journeys would not be the same 30 journeys, so its mean would probably differ from the first. This variation is called sampling variability.

Sampling variability does not mean that someone made a mistake. It means that a sample is only a part of the population. A larger sample usually varies less from sample to sample, so it gives a more reliable estimate.

Example 4. Two samples of size 20 are taken from the same population. Their means are 24 minutes and 27 minutes. What is the best estimate of the population mean from these two samples?

Both samples are the same size, so the mean of the two sample means is a sensible estimate:

\frac{24 + 27}{2} = 25.5 \text{ minutes}.

The two samples disagree by 3 minutes, which shows the sampling variability.

Example 5. A sample of five values is $4, 5, 6, 7, 8$ . Another sample of five is $4, 5, 6, 7, 30$ . Compare the two means.

For the first sample,

\text{mean} = \frac{4 + 5 + 6 + 7 + 8}{5} = \frac{30}{5} = 6.

For the second sample,

\text{mean} = \frac{4 + 5 + 6 + 7 + 30}{5} = \frac{52}{5} = 10.4.

One unusual value raises the mean by $4.4$ . Section 11.6 returns to values like this one.

6. Try it: how much does a sample mean vary?

Predict first. The figure shows a population and draws a random sample from it. Before you touch a control, predict what happens to the sample mean when you make the sample larger. Write down whether the sample means will spread out more or less, and why.

Test. Set the sample size to its smallest value and take several samples. Watch where each sample mean lands. Now set the sample size to its largest value and take several more samples.

Explain. Write two sentences. The first explains why two samples of the same size give different means. The second explains why a larger sample usually gives a mean closer to the population mean.

Watch out. Three mistakes are common.

Treating the sample as the population. A conclusion about the 30 timed journeys is not yet a conclusion about every journey. Say which one you mean.

Rounding an estimate too early. Keep the value on your calculator until the last line. Rounding $0.06$ to $0.1$ before multiplying would give a different estimate.

Believing that a bigger sample removes all error. A large sample still varies, and a large biased sample is still biased. Section 11.2 shows how a poor method can ruin a large sample.

Re-check: after any estimate, ask whether it is a count, a proportion or a total, and whether the units match the question.

You should now be able to

explain the terms population, sample, census and sampling frame;
give two reasons why a sample is used instead of a census;
estimate a population total from a sample mean;
estimate a population count or proportion from a sample;
explain why two samples from one population can disagree;
state whether a conclusion describes the sample or the population.