6.4 - Random samples and unbiased estimates
You rarely measure every member of a population, so a sample has to support conclusions about a larger group. In this first part of 6.4, you will distinguish parameters from statistics, choose samples with random numbers, recognise sources of bias and calculate unbiased estimates of a population mean and variance.
Population and sample
Would asking a thousand library visitors about the whole school's reading habits be better than randomly choosing a hundred students? The larger count does not repair the first survey's missing students: people who never visit the library cannot be selected there.
The group you want to describe must be defined before you choose observations. “All students enrolled at this school this term” is a population. “Everyone who answers our website poll” is a group of respondents, which may differ systematically from the intended population.
Population, sample, parameter and statistic
A population is the complete group of individuals or measurements of interest. A sample is a selected subset of observations from that population.
A parameter, such as population mean or variance , describes the population. A statistic, such as sample mean , is calculated from a sample. The parameter is fixed for the defined population; the statistic can change from sample to sample.
For example, suppose a random sample of bottles from one day's production has mean volume ml. That is a statistic. The day's population mean is an unknown parameter, not automatically . Measuring a different random sample can give a different mean even when both samples are chosen properly.
Random selection gives members a defined chance of inclusion and prevents a researcher deliberately choosing convenient or preferred observations. A simple random sample of fixed size gives every subset of that size the same chance of selection. Choosing uniformly from a complete numbered list without replacement achieves this.
Randomness does not guarantee that one sample perfectly matches its population. It controls the selection process so that sampling variation can be described by probability. A larger well-chosen sample usually reduces that variation; it does not make a systematically excluded group appear.
Key idea
Define the population first, then use a selection process that can reach it. A sample statistic estimates a population parameter; it does not become the parameter merely because the sample is large.
Check which quantity changes when the sample changes.
Unsatisfactory sampling
A sampling method is biased if its selection tends to favour particular types of observation in a way that distorts the quantity being studied. Explain the link to the context, rather than simply calling a sample “unrepresentative”.
For a survey of the whole school's reading habits, interviewing library visitors overrepresents students who use the library and excludes non-visitors. A voluntary reading poll may attract students with particularly strong interests. Taking the first names on an alphabetical list avoids personal choice but is not random selection: many other students have no chance of inclusion.
Use a complete list and random numbers to avoid those selection rules. Label every member once using the same number of digits. Read equally likely digit groups in a fixed order, reject unused labels, and reject repeats when sampling without replacement. Continue until the required number of distinct members has been selected. Decide these rules before looking at the generated numbers.
Worked example 1: two-digit labels
Question. Choose a random sample of people from , using the digit groups
.
Describe the procedure and identify the selected labels.
Plan. Label people to , accept valid groups in order and reject repeats.
| Group | Decision | Reason |
|---|---|---|
| Accept | In range and new | |
| Reject | Above | |
| Accept | In range and new | |
| Reject | Already selected | |
| Reject | No person has this label | |
| Accept | In range and new | |
| Accept | Fourth distinct selection |
Answer. Select people , then stop.
Check. There are four distinct labels, all between and . Every valid two-digit label has the same chance of appearing under the uniform random-number process.
Increase only the population size. Three-digit groups are now needed to label every person.
Worked example 2: three-digit labels
Question. A population has people. Describe how to choose using random numbers, and apply your rule to
.
Plan. Label people to and apply the same acceptance rule to three-digit groups.
Accept , reject , accept , reject the repeated , reject , then accept and .
Answer. Generate uniform three-digit random numbers, ignore , values above , and repeats. Select .
Check. Each chosen label occurs once and lies in the population. Using only two digits would have excluded everyone labelled or more, so the change in digit length matters.
Dividing a random number by the population size and taking a remainder can give unequal selection chances unless the original range divides exactly into equally sized groups. The rejection procedure above avoids that problem. Random numbers also cannot repair an incomplete list: if the list omits non-library users, random selection from that list still misses them.
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