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Select a fixed number of distinct units fairly, handle repeated random labels, and understand what makes a sample simple random. Original examples and worked practice.
Before you startPopulations, sampling frames and elementary probability.
01 / Every sample gets the same chance
For a fixed size n, every possible n-unit subset has the same probability of selection.
Here we sample distinct units without replacement.
From four labelled units A, B, C and D, there are six samples of size two: AB, AC, AD, BC, BD and CD. If each has chance 1/6, the selection is simple random.
Compare the two rules in the model. Equal selection chances for individuals alone are not sufficient to make a design simple random.
Each of the six samples has probability 1/6: this is simple random sampling.
Every individual has inclusion probability 1/2 under both designs.
For example, A appears in AB, AC and AD: 3 × 1/6 = 1/2.
The control compares probability rules; it does not make a random draw. Pair order does not matter: AB and BA are the same sample.
02 / Start with a complete frame
Check the list contains all 72 eligible volunteers once
Remove duplicates and ineligible entries before sampling.
Assign labels 01 to 72
Each label refers to exactly one volunteer.
Choose 8 distinct labels by a fair random method
Every subset of eight should have the same chance.
Contact the volunteers attached to those labels
Retain a record of the selected labels and any nonresponse.
A researcher wants 6 of 48 current club members. Describe the first step before generating numbers.
Numbers must correspond to actual eligible units.
Make or check a current list containing all 48 members once and assign unique labels 01–48.
03 / Use random integers
Generate independent uniformly distributed integers from 1 to N, inclusive. Keep a number only if it has not already been selected. Continue until n distinct labels have been accepted. A calculator or program must be configured for the full required range; “pick random numbers” alone leaves important details unspecified.
Accept 9 and 3
They are in range and new.
Reject the second 9
It would select the same unit twice.
Accept 12 and 1, then stop
The sample has four distinct units: 9, 3, 12 and 1. The later 6 is not needed.
For a sample of five from labels 1–20, the sequence is 7, 7, 18, 2, 18, 11, 20, 5. Which labels are accepted?
Ignore repeated labels and stop at five distinct ones.
7, 18, 2, 11, 20. Reject the second 7 and second 18. Stop before 5.
04 / Use a random-digit table
Use labels 01–64
The generator may produce 00–99.
Read in a direction chosen in advance
Do not search for preferred numbers.
For 83, 04, 00, 61, 04, 27, accept 04, 61, 27
Reject out-of-range groups and repeated accepted labels.
Stop at three distinct valid labels
Rejection preserves equal chances among remaining valid labels.
For labels 01–35, process 36, 00, 35, 08, 08, 19 to select three units.
35 is valid; 36 and 00 are not.
Accept 35, 08 and 19. Reject 36, 00 and the repeated 08.
For 64 units, someone maps random integers 1–100 to labels by subtracting 64 whenever the result is above 64. What goes wrong?
Compare how many source integers map to labels 1 and 64.
Labels 1–36 each have two possible source integers, whereas labels 37–64 each have only one. Selection is unequal. Reject 65–100 instead, or generate uniform integers directly from 1–64.
05 / Use a lottery
Write every unique label on an indistinguishable ticket, mix thoroughly, draw without looking and do not replace selected tickets. Draw the required number. Different ticket sizes, incomplete mixing or choosing visible labels could defeat the intended fairness.
Why leave drawn tickets out when selecting ten distinct people?
The required units must be distinct.
Leaving them out prevents the same person being selected twice and gives sampling without replacement.
Give a practical limitation of a ticket lottery for 200,000 units.
The method can be fair but inconvenient.
Preparing, storing and thoroughly mixing 200,000 indistinguishable tickets is impractical; a suitable random-number method is easier.
06 / Equal individual chances
A, B, C and D each have inclusion probability 1/2
Each appears on exactly one coin outcome.
AB and CD each have probability 1/2
The other four pairs cannot occur.
This is not a simple random sample of size two
All six pairs would need the same chance.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
“Every person has an equal chance, therefore it must be simple random sampling.” Is this sufficient?
Use the coin example.
No. Equal individual inclusion probabilities are necessary for a fixed-size simple random sample, but not sufficient. The AB/CD design is a counterexample.
07 / Count small sample spaces
From A, B, C, D, drawing two distinct units in order gives 4 × 3 = 12 sequences. Each unordered pair occurs in two orders, so there are 12 ÷ 2 = 6 samples. Under a fair draw without replacement each sequence has probability 1/12, so each pair has probability 2/12 = 1/6.
Under the six-pair simple random design, find the probability that C is included.
List the pairs containing C.
AC, BC and CD: 3/6 = 1/2.
How many simple random samples of size one are possible from five units, and what is each sample’s probability?
Each singleton is a different sample.
Five samples, each with probability 1/5.
08 / Random does not mean perfect
A simple random method limits deliberate selection preferences and has a clear probability structure. It does not guarantee every small group appears in one particular sample. It needs a suitable frame, and nonresponse or poor measurement can still distort the collected data.
A small random sample happens to contain no members from one small department. Does this prove the random generator is faulty?
A possible sample can be unbalanced by chance.
No. Simple random sampling does not guarantee every subgroup is represented in each realised sample. Stratified sampling can ensure a planned allocation across departments.
A selected person does not reply. Why is replacing them with a friend of the interviewer problematic?
Was the replacement chosen by the stated random rule?
No. Convenience replacement breaks the planned design and may introduce bias. Use a preplanned defensible follow-up or replacement procedure and report nonresponse.
09 / Make the procedure reproducible
Describe how to take a simple random sample of 12 from 150 current members using a calculator.
Include enough detail for someone else to repeat the procedure.
Check a complete, deduplicated current membership list. Label members 1–150. Generate independent uniform random integers from 1 to 150 inclusive. Ignore previously accepted labels and continue until 12 distinct numbers are accepted. Select the members with those labels; record and handle nonresponse under a stated plan.
Section 1 of 9 · Every sample gets the same chance