S1, S5 - Samples, populations and statistical inference

S1, S5 - Samples, populations and statistical inference

How can a small survey say anything about a much larger group? S1 explains how samples support an inference and why their selection matters. S5 uses those results to describe a population, with a clear distinction between an observed count and an estimate.

Populations, samples and representativeness

Can asking 80 pupils tell a school how its 900 pupils travel? It can provide evidence, but first you must be clear about whom the school wants to describe.

You already know

A proportion compares a part with its whole. If 29 of 80 pupils use a bus, their observed proportion is 2980=0.3625\frac{29}{80}=0.3625.

The 80 selected pupils are the group observed. The 900 pupils are the group the school wants to learn about. The 29 bus users are one category within the sample, not the sample itself.

Population, sample and census

The population is the complete group about which a statistical question is asked. A sample is a smaller group selected from it. A census collects information from every member of that population.

The boundary matters. A club randomly selects 45 people from its 360 current adult members. Those 360 adults are the population, and the 45 selected adults are the sample. Children and former members are outside this population; the result cannot automatically describe all local residents.

A representative sample reflects the relevant features of its population well enough for the intended inference. If year groups have different travel patterns, selecting only one year could misrepresent the school. A sample need not contain every possible kind of pupil, but its selection should give a fair picture of features relevant to the question.

Key idea

Define the target group before judging the sample. Measuring every tree in one orchard is a census of that orchard, even though other orchards exist.

Representativeness concerns who is reached and how they are selected. A large headcount alone cannot establish it.

Bias, variation and sample size

A sample can mislead because the selection systematically favours a group, or simply because a fair sample varies by chance.

Bias

Bias is a systematic tendency for the evidence to favour some members or outcomes. Asking about school journeys only at the cycle racks overrepresents cyclists and misses many walkers and bus users.

Two fair random samples of 80 pupils can contain different numbers of bus users. This is sampling variation: chance changes which people are included. A larger unbiased sample generally gives a more stable estimate, but does not guarantee the exact population proportion.

Bias needs a change to the procedure. Asking three times as many people at the cycle racks still favours cyclists. Even 2000 volunteer replies to an online library poll may give a less representative picture than a smaller random member sample: members without access to the poll have no opportunity to reply.

Also examine the sampling frame, the list or route used to reach the population. A complete random selection from a list missing one year group still misses that group. Non-response matters too: if pupils with long journeys reply less often, their journeys become underrepresented among the respondents. A vague question such as “Do you visit often?” can mean different things to different people. A defined month and a count of visits produce more comparable answers.

Key idea

Random selection reduces selection bias; it does not remove chance variation or guarantee that everyone selected replies. More responses reduce neither a missing-group problem nor misleading wording automatically.

Describe the affected group and its possible effect. If no evidence tells you which group has higher values, say that the estimate may be unrepresentative rather than inventing a direction of error.

Choosing a fair sampling procedure

A fair procedure gives the intended population a fair opportunity to be selected and says exactly how selection is made.

For a simple random sample, start with a complete list, give every member a unique number and select distinct numbers uniformly at random. Reject repeats when you need different people. “Pick some people at random” is too vague to describe a reproducible procedure.

Worked example 1: selecting distinct students

Question. Describe a simple random sample of 25 different students from a complete list of 480.

Plan. Number the whole list and select without repeating a student.

  1. Assign identifiers 11 to 480480: include everyone.
  2. Generate uniform random integers from 11 to 480480: give each identifier the same chance.
  3. Reject repeats until there are 2525 distinct identifiers: avoid counting anyone twice.
frame=1,2,…,480cover the populationsample count=25 distinct identifiersstop at the required size\begin{aligned} \text{frame}&=1,2,\ldots,480 &&\text{cover the population}\\ \text{sample count}&=25\text{ distinct identifiers} &&\text{stop at the required size} \end{aligned}

Why this method? Alphabetical position or availability must not decide who enters the sample.

Answer. Contact the 25 students with the selected identifiers.

Check. There are 25 different selections, all drawn from the complete 480-person list.

The next example changes only the required sample size. The selection rule remains the same.

Worked example 2: a larger random sample

Question. Describe a simple random sample of 40 different students from that same list of 480.

Plan. Keep the complete numbered frame and change the stopping count.

  1. Keep identifiers 11 to 480480: the population is unchanged.
  2. Generate uniform random identifiers, rejecting repeats: retain equal selection chances.
  3. Stop at 4040 distinct identifiers: match the requested sample size.
frame=1,2,…,480same complete populationsample count=40 distinct identifiersnew stopping count\begin{aligned} \text{frame}&=1,2,\ldots,480 &&\text{same complete population}\\ \text{sample count}&=40\text{ distinct identifiers} &&\text{new stopping count} \end{aligned}

Why this step? Increasing the sample must not mean taking the next 15 names alphabetically.

Answer. Survey the 40 selected students.

Check. Every selected identifier is valid and no student appears twice.

A systematic sample instead selects at a fixed interval after a random start. Check that the list has no repeating pattern matching that interval.

Worked example 3: systematic selection

Question. Select 24 people systematically from an ordered list of 240.

Plan. Divide population size by sample size, then choose a random start within one interval.

interval=240÷24=10space selections equallypositions=7,17,27,…,237if the random start is 7count=(237−7)÷10+1=24include both endpoints\begin{aligned} \text{interval}&=240\div24=10 &&\text{space selections equally}\\ \text{positions}&=7,17,27,\ldots,237 &&\text{if the random start is 7}\\ \text{count}&=(237-7)\div10+1=24 &&\text{include both endpoints} \end{aligned}

Why this step? Choosing the start uniformly from 1 to 10 avoids always starting with the first member.

Answer. Select those 24 positions, provided the list has no relevant ten-entry repeating pattern.

Check. The final position is within the list and there are exactly 24 selections.

Where relevant groups differ, a plan can represent them in sensible proportions, with fair selection within each group. Convenience sampling is easier but provides weaker evidence about everyone. Use neutral questions, defined periods and units; record non-response rather than quietly treating respondents as the entire intended sample.

Watch out

Wrong: “Number only the pupils present in the library, then choose randomly.” Random selection from an incomplete frame still excludes other pupils. Use the complete enrolment list before choosing distinct identifiers.

Now change only the population size in the random-selection method.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account