Understand · explore · practise
Plan a sampling investigation, calculate estimates from an original small dataset and judge the design. Includes a learner-operated comparison and downloadable spreadsheet.
Before you startSampling methods, means and data types.
01 / Turn a question into a plan
Question → population → variable → frame → selection → analysis → conclusion.
Write the design before looking for a result you prefer.
Our invented library dataset contains twenty enquiries: twelve in a morning session and eight in an evening session. We will estimate their mean duration using a sample of five, then compare it with the known full-population mean.
The complete values are available for learning, so we can check estimation error. In a real investigation, the population result is usually unknown.
Selected IDs: 1, 2, 3, 4, 5.
38 minutes ÷ 5 records = 7.6 minutes.
Opportunity selection from the start of the list: all five are morning records.
All data are invented for this lesson. Closeness in one sample does not prove that a selection method is reliable.
02 / Define the target and measurement
The target population for this exercise is the twenty invented enquiries listed here. The unit is one enquiry, the variable is its duration in minutes and the session is a grouping variable. Duration is an underlying continuous measurement even though the teaching values are whole numbers.
For a real study, specify the library, dates, eligible enquiry types and how duration starts and ends. An enquiry spanning closing time needs a stated handling rule, not an improvised decision.
Why is “the library” not the sampling unit in this exercise?
What object is selected and measured each time?
We select individual enquiries. Each enquiry supplies one duration, so the enquiry is the sampling unit.
Would a census of these twenty invented enquiries establish the mean for all libraries?
The source and scope are much narrower.
No. These are synthetic teaching records, and even a real one-library period would not automatically represent all libraries.
03 / Choose and justify a design
Morning allocation = 5 × 12/20 = 3
The morning group is 60% of the population.
Evening allocation = 5 × 8/20 = 2
The evening group is 40%.
Choose three distinct IDs from 1–12 and two from 13–20 randomly
The allocation alone is not a random selection method.
A simple random sample of five from all twenty is also a legitimate design, but its group counts vary. Selecting the first five records is convenient and excludes evening records completely.
Why would two morning and three evening selections not be proportional here?
Compare sample shares with population shares.
The sample would be 40% morning and 60% evening, reversing the population’s 60%/40% split.
A learner chooses the three shortest morning enquiries and two shortest evening enquiries. Is that stratified random sampling?
The selection uses the outcome values.
No. Choosing the shortest is deliberately selective within each stratum. Select by a fair random process on IDs, not by preferred durations.
04 / Record the actual selected IDs
State the method and record the resulting IDs. If generating integers with replacement, reject duplicate or out-of-range labels. Do not redraw simply because the first result looks inconvenient.
The model contains fixed illustrative selections. The “random outcome” labels mean these sets could result from the stated random designs; choosing the menu option is not itself a random draw.
Random labels for a sample of five arrive as 2, 2, 15, 9, 21, 12, 5. The valid IDs are 1–20. Which five distinct valid IDs are accepted?
Reject the repeated 2 and the out-of-range 21.
2, 15, 9, 12 and 5.
Why should you not keep redrawing until the sample mean is close to the population mean?
You would be selecting outcomes based on the answer.
That changes the procedure and hides sampling variation. Keep and report the result from the planned selection, even if its estimate is less accurate.
05 / Try the spreadsheet investigation
Download the sampling investigation workbook (.xlsx). It contains only the original invented data, selection cells and transparent formulas. Set Include to 1 for selected rows and 0 for the others.
You can also use the table below without a spreadsheet. In the workbook, the formulas calculate from your 0/1 selection. A zero selection shows “No selection” instead of dividing by zero.
Invented data: library enquiry durations.
| ID | Session | Minutes |
|---|---|---|
| 1 | Morning | 5 |
| 2 | Morning | 7 |
| 3 | Morning | 9 |
| 4 | Morning | 11 |
| 5 | Morning | 6 |
| 6 | Morning | 8 |
| 7 | Morning | 10 |
| 8 | Morning | 12 |
| 9 | Morning | 7 |
| 10 | Morning | 9 |
| 11 | Morning | 11 |
| 12 | Morning | 13 |
| 13 | Evening | 10 |
| 14 | Evening | 12 |
| 15 | Evening | 14 |
| 16 | Evening | 16 |
| 17 | Evening | 11 |
| 18 | Evening | 13 |
| 19 | Evening | 15 |
| 20 | Evening | 17 |
For IDs 2, 5, 9, 12 and 15, calculate the mean duration.
The durations are 7, 6, 7, 13 and 14 minutes.
Total 47 minutes; mean 47/5 = 9.4 minutes.
For IDs 2, 6, 10, 15 and 18, calculate the mean and its difference from 10.8 minutes.
The durations are 7, 8, 9, 14 and 13.
Total 51; mean 51/5 = 10.2 minutes. Sample mean minus population mean = −0.6 minutes.
06 / Compare results without overclaiming
Different samples give different estimates. The first five records average 7.6 minutes; our illustrative simple random outcome gives 9.4 and our stratified outcome gives 10.2. In these particular examples the stratified estimate is closer to 10.8, but that does not prove that every stratified sample beats every simple random sample.
Repeated samples help reveal how a design behaves. Keep the sampling rule fixed and examine all results rather than displaying only the closest estimate.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
An estimate is 9.4 minutes and the known population mean is 10.8. Give its signed difference and absolute error.
Subtract estimate minus true value, then take the magnitude.
Signed difference −1.4 minutes; absolute error 1.4 minutes.
07 / Plan for practical problems
In a real enquiry study, a missing end time means the duration is unknown. A zero-duration replacement is not justified. State the handling policy and report the usable sample size. Record durations consistently and keep identifiers with their measurements when sorting.
If human participants are involved in an actual investigation, follow your school’s or institution’s privacy and consent arrangements. This exercise uses no real personal records.
Five selected records include one unknown duration. What should the report say if the mean uses the other four?
Distinguish selected and usable records.
Five were selected but only four known durations contributed. Explain the missing-data rule and note that the four may not represent the missing enquiry.
Why is sorting the duration column alone, while leaving IDs and sessions fixed, damaging?
A row’s fields belong together.
It assigns durations to the wrong enquiries and sessions. Sort the whole table so each record stays intact.
08 / Write a defensible conclusion
A useful report names the population, method, selected IDs, usable count, numerical estimate and limitations. The answer should match the original question.
Write a short conclusion for IDs 2, 6, 10, 15 and 18.
Include the allocation, mean and scope.
This illustrative proportional stratified sample contains three morning and two evening enquiries. Its five durations total 51 minutes, giving a mean of 10.2 minutes, 0.6 below the known twenty-record mean. The data are invented; this single outcome does not establish that the method always gives the closest estimate.
Section 1 of 8 · Turn a question into a plan