Lesson 1.11 · Unit 1 · CED topic 1.11
Random sampling methods
A conclusion about a population is only as good as the sample it comes from. The rest of the course depends on one idea: if the sample is chosen with a random mechanism, then probability tells us how far the sample is likely to be from the population, and inference is possible. This lesson describes the four standard random sampling methods, shows how to choose a simple random sample, and explains how to decide which method fits a given situation.
Learning objectives.
- 1.11.A. Identify a sampling method given a description of a study.
- 1.11.B. Justify the appropriateness of a sampling method.
Essential knowledge.
- 1.11.A.1. Sampling without replacement is a sampling strategy in which an observational unit from a population can be selected only once. The observational unit is not returned to the population before subsequent selections of observational units are made, so there is no chance that the observational unit can be selected again.
- 1.11.A.2. Sampling with replacement is a sampling strategy in which an observational unit from the population can be selected more than once. The observational unit is returned to the population before subsequent selections of observational units are made, so it is possible that the observational unit could be selected again.
- 1.11.A.3. In a simple random sample (SRS) of size n, every sample of the size n has the same chance of being selected. This method is the basis for many types of sampling mechanisms. There are several procedures to obtain a simple random sample; for example, using a random number generator or randomly selecting numbered slips of paper.
- 1.11.A.4. A stratified random sample involves the division of all individuals in a population into non-overlapping groups, called strata, based on one or more shared attributes or characteristics (homogeneous grouping). Within each stratum a simple random sample is selected, and the selected individuals are combined to form one sample.
- 1.11.A.5. A cluster random sample involves the division of a population into smaller groups, called clusters. Ideally, each cluster mirrors the heterogeneity of the population, with clusters similar to one another. A simple random sample of clusters is selected from the population to form the sample of clusters. Data are collected from all observational units in each of the selected clusters.
- 1.11.A.6. A systematic random sample is a method in which sample members from a population are selected according to a random starting point and a fixed, periodic interval between successive sampling units.
- 1.11.B.1. Each random sampling method has different characteristics that make it more appropriate for sampling populations depending on the question being investigated.
With and without replacement
Sampling without replacement means an observational unit can be selected only once: after a unit is chosen it is not returned to the population before the next selection, so there is no chance it can be chosen again. Sampling with replacement returns each selected unit to the population, so it is possible for the same unit to be selected more than once. Most real surveys sample without replacement (you do not interview the same person twice). When the sample is a small part of the population, the two ways behave almost identically, which is the reason for the "10% condition" you will meet in inference.
Simple random sample
In a simple random sample (SRS) of size \(n\), every sample of size \(n\) has the same chance of being selected. It is the basis for the other methods. Two common ways to choose one: write each unit's number on a slip of paper, mix thoroughly and draw \(n\) slips; or number the units and use a random number generator. On the TI-84, `randIntNoRep(1, N, n)` (MATH, PRB, 8:randIntNoRep) returns \(n\) different integers from 1 to \(N\) in random order; on a computer, a random sample function does the same.
Stratified random sample
A stratified random sample divides all the individuals in the population into non-overlapping groups called strata, based on one or more shared attributes or characteristics, so that the individuals in each stratum are alike (homogeneous grouping). Within each stratum a simple random sample is selected, and the selected individuals are combined into one sample. Strata might be grade levels, age groups, or regions. The strata should be groups that differ from one another on the variable of interest.
Cluster random sample
A cluster random sample divides the population into smaller groups called clusters. Ideally each cluster mirrors the heterogeneity of the whole population, so the clusters are similar to one another. A simple random sample of clusters is selected, and data are collected from every observational unit in each selected cluster. Clusters might be city blocks, classrooms or hospitals.
Systematic random sample
A systematic random sample selects members according to a random starting point and a fixed, periodic interval between successive units: for example, a random start and then every 15th unit.
The easiest way to keep the methods apart is to ask what is random and what is certain. Stratified: every stratum is represented, with a random sample from each. Cluster: only some clusters are represented, but everyone within a chosen cluster is surveyed.
Choosing a method
Each random sampling method has characteristics that make it more appropriate for sampling certain populations, depending on the question being investigated.
| Method | A good fit when... |
|---|---|
| Simple random sample | a complete list of the population exists and the population is fairly uniform; a simple baseline |
| Stratified | the population has groups that differ on the variable of interest (grades, regions), and you want each group represented and a more precise estimate |
| Cluster | the units come in natural groups that are alike one another, and it is cheaper or easier to survey whole groups (blocks, classrooms) |
| Systematic | the units come in an ordered list or line (an assembly line, a roster) and a random start with a fixed interval is easy to carry out |
A systematic sample is not an SRS: after the random start the sample is fixed, so some groups of \(n\) units cannot occur together. Still, it is considered random because the start is chosen randomly.
A club has 200 members, numbered 1 to 200. Choose a simple random sample of 5. The calculator command `randIntNoRep(1,200,5)` returns 137, 24, 188, 92, 5. Which members are selected?
Members numbered 5, 24, 92, 137 and 188 are in the sample. Since the command returns distinct numbers, no member is chosen twice (sampling without replacement), and every group of 5 members had the same chance of being selected, which is what makes it a simple random sample. If a repeated number had appeared with a method that samples with replacement, you would simply discard it and draw again when you need distinct people.
Identify the method used in each study. (a) A school lists students by grade (9, 10, 11, 12) and randomly selects 15 from each grade. (b) A researcher randomly selects 8 of a city's 40 blocks and surveys every household on those 8 blocks. (c) An inspector picks a random start among the first 10 items on an assembly line and then tests every 10th item. (d) All 300 employees are numbered and a random number generator picks 40.
(a) Stratified: the grades are the strata and an SRS is chosen in each. (b) Cluster: the blocks are the clusters, a random sample of clusters is chosen, and all households in them are surveyed. (c) Systematic: a random start and a fixed interval of 10. (d) Simple random sample: every set of 40 employees is equally likely.
A list has \(N=600\) customers and a sample of \(n=40\) is wanted. Describe a systematic sample.
The interval is \(k=N/n=600/40=15\). Choose a random starting number from 1 to 15, say 7, and then select every 15th customer: 7, 22, 37, 52, and so on, up to \(7+39(15)=592\), giving exactly 40 customers. Every customer had a chance \(1/15\) of being the one chosen in his or her group of 15, but only 15 different samples are possible, so it is not a simple random sample.
A county has five towns whose household incomes differ greatly. The county wants to estimate the mean household income. Why might a stratified sample be better than a simple random sample?
A simple random sample could, by chance, include too many households from the richest or the poorest town and give a poor estimate. If the towns are the strata, the sample guarantees that every town is represented, and since the households within a town have similar incomes, the estimates from the strata vary less. Stratifying works best when the strata differ on the variable being studied and the units within each stratum are alike.
- SRS: every sample of size \(n\) is equally likely. Stratified: SRS within each stratum (homogeneous groups). Cluster: SRS of clusters, then all units in each chosen cluster (clusters are alike each other). Systematic: random start, then a fixed interval \(k=N/n\).
- Use a random mechanism (random number generator, `randIntNoRep`) to choose the units, and say how.
- Justify the method with the situation: stratify when groups differ on the variable; cluster when groups are alike and whole groups are convenient; systematic for ordered lists or lines.
- Sampling without replacement means a unit can be chosen only once.
- A high school has 1,000 students: 400 freshmen, 300 sophomores, 200 juniors and 100 seniors. A stratified sample of 50 students uses grades as strata and takes a number proportional to the size of each grade. How many students come from each grade?
Show answer
The sample is 5% of the population: freshmen \(0.05(400)=20\), sophomores 15, juniors 10, seniors 5 (a total of 50). Within each grade an SRS of that size is selected.
- Explain the difference between a stratified random sample and a cluster random sample.
Show answer
In a stratified sample the population is divided into groups that are alike inside (strata) and a random sample is taken from every stratum. In a cluster sample the population is divided into groups that each resemble the whole population (clusters), a random sample of some clusters is chosen, and everyone in the chosen clusters is surveyed.
- A list of 900 voters is to be sampled with \(n=30\) using a systematic sample. What is the interval and how is the start chosen?
Show answer
The interval is \(k=900/30=30\). Choose a random start from 1 to 30 and then take every 30th voter on the list.
- A teacher says: "I stood at the door and picked the first 10 students who came in, so I have a random sample." Is the teacher right? Why or why not?
Show answer
No. The first 10 students are a convenience sample, chosen by arrival order rather than by a random mechanism, so the method is not random and the sample may not represent the whole class (for example, early arrivals may be more punctual or may differ in other ways).
- Describe how to use a calculator to select a simple random sample of 6 students from a class of 80 numbered 1 to 80.
Show answer
Number the students 1 to 80, then use `randIntNoRep(1,80,6)` to generate six different integers. The students with those numbers form the sample. (If you use `randInt` and a number repeats, discard it and draw again.)
- A researcher wants to study the diet of the people living in a city that has 60 neighborhoods, each similar to the others in income and age. Interviewing everyone in a few randomly chosen neighborhoods would be cheaper than traveling across the whole city. Which method is this and why is it reasonable?
Show answer
This is a cluster random sample. It is reasonable because the neighborhoods (clusters) are alike one another and each reflects the whole city, and it is cheaper to survey all of the people in a few clusters.