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eduKateSG · Why Mathematics? · Finite Population Correction: Sampling Fractions and Precision Without Replacement
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a sample covers a large share of a school, neighbourhood or business population? Sampling without replacement removes units from the pool. The finite population correction measures the resulting reduction in uncertainty, so precision calculations do not pretend sampled units could be selected again.
Did You Know? If every unit in a finite population is observed without replacement, sampling variance is zero. The correction falls naturally to zero at a census.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
For a tiny sample from a huge population, removing one unit barely changes the next draw. When n is a substantial part of N, successive draws contain progressively less unknown population.
Under simple random sampling without replacement, Var(sample mean)=(1-f)S²/n, where f=n/N. Standard errors use sqrt(1-f).
Statistics Canada notes that selecting 50 of 135 schools gives sampling fraction 0.37 and requires a finite population correction. Read the official simulation section.
Suppose N=100 students and n=40 are sampled without replacement. The sampling fraction is f=40/100=0.40.
| Population N | Sample n | Fraction f |
|---|---|---|
| 100 | 40 | 0.4 |
The finite population variance multiplier is 1-f=0.60.
CHAPTER 6 OF 34 · Understand the question
6. Calculate the standard-error multiplier
The standard error is multiplied by sqrt(0.60)≈0.7746 compared with a with-replacement calculation using the same S²/n component.
| Variance multiplier | Standard-error multiplier |
|---|---|
| 0.6 | 0.7746 |
Sampling 40 distinct students reveals information about 40% of the finite list. Only 60% remains outside the sample, so uncertainty is smaller than repeated independent draws would imply.
When n=1, the factor is close to one for large N. When n=N, 1-f=0 and the sample mean equals the finite-population mean exactly.
The usual correction does not apply to independent with-replacement draws because an observed unit can reappear and the pool is not depleted.
N is the number of eligible units on the target frame at the selection stage. Using national population when sampling within one school destroys the meaning of f.
In multistage designs, a correction belongs to the stage where sampling without replacement occurs. A high student fraction inside schools does not replace the first-stage school correction.
Compute f_h=n_h/N_h inside each stratum. One overall n/N can miss large fractions in small strata.
The independent units for a stage may be schools or classes, not students. Applying a student-level FPC to a small number of selected clusters can severely understate uncertainty.
Some files suppress stratum population sizes or sampling fractions. Analysts should follow supplied replicate weights and documentation instead of inventing an FPC.
For a desired margin of error, a finite-population formula can reduce the required n when the target population is small. The adjustment is justified by the actual without-replacement design.
The FPC can reduce variance; clustering and unequal weights can increase it; stratification may reduce it. Total precision reflects all components together.
Sampling fraction describes selection from a frame. Response rate describes participation among eligible selections. A high sampling fraction does not excuse nonresponse bias.
Verify N, n, sampling stage, replacement rule, strata, certainty units and whether replicate weights already incorporate the correction.
Reproduce 0.40, 0.60 and 0.7746; enumerate small samples without replacement; compare empirical variance with and without the correction; and test n=N.
The FPC does not correct frame error, nonresponse, measurement error, clustering or a wrong target population. It only addresses depletion under the specified probability design.
Rules such as “ignore below five or ten percent” are approximations. Use exact design information when available, especially for high-stakes or multi-stage estimates.
Store N, n, f, 1-f, sqrt(1-f), sample variance and the corrected standard error. Keep stage and stratum labels visible.
Enumerate all fixed-size samples from a tiny population, calculate mean variance and compare it with (1-f)S²/n.
Survey statisticians, education researchers, auditors and quality engineers use finite-population corrections.
Sample 4 of 10 known values, enumerate every combination and show the variance falling as n approaches 10.
A large response rate is not an FPC, sample fraction is not outcome prevalence, and the correction cannot be attached to the wrong sampling stage.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The design-effect guide combines design features. The Horvitz–Thompson guide explains inclusion probabilities.
Students should name the finite list and the without-replacement stage. Parents can ask whether a school survey sampled pupils, classes or schools as the independent units.
- Calculate f.
- Calculate both multipliers.
- Explain n=N.
- Apply an FPC by stratum.
- Reject a wrong-stage correction.
### Is the FPC always needed? Only when the design and sampling fraction make it relevant.
### Does it change the point estimate? Usually no.
### Is it 1-f or sqrt(1-f)? Variance uses 1-f; standard error uses its square root.
### Does it fix bias? No.
Report the target frame, N, n, f by stage and stratum, replacement rule, estimator, variance method, replicate treatment and nonsampling-error limits.
Without replacement, every selected unit reduces the unobserved finite set. The correction quantifies this negative dependence.
Confirm the eligible population and stage, compute exact fractions, use the variance or standard-error form correctly, avoid double correction and keep nonsampling errors separate.
Continue through the Mathematics Learning Hub. Recalculate one design quantity and explain which source of uncertainty it represents.
Extended practice laboratory
Practice case 1: Define frame
Setup: Start from the article’s finite-population-correction mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Set N
Setup: Start from the article’s finite-population-correction mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Set n
Setup: Start from the article’s finite-population-correction mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Calculate f
Setup: Start from the article’s finite-population-correction mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate 1-f
Setup: Start from the article’s finite-population-correction mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Calculate square root
Setup: Start from the article’s finite-population-correction mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Correct variance
Setup: Start from the article’s finite-population-correction mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Correct standard error
Setup: Start from the article’s finite-population-correction mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Set n=N
Setup: Start from the article’s finite-population-correction mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Use small n
Setup: Start from the article’s finite-population-correction mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Add stratum
Setup: Start from the article’s finite-population-correction mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Add second stratum
Setup: Start from the article’s finite-population-correction mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Add cluster stage
Setup: Start from the article’s finite-population-correction mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Add student stage
Setup: Start from the article’s finite-population-correction mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Check replacement
Setup: Start from the article’s finite-population-correction mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Check replicate weights
Setup: Start from the article’s finite-population-correction mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Compare response rate
Setup: Start from the article’s finite-population-correction mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Compare design effect
Setup: Start from the article’s finite-population-correction mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Change N
Setup: Start from the article’s finite-population-correction mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Change n
Setup: Start from the article’s finite-population-correction mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Audit frame
Setup: Start from the article’s finite-population-correction mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Audit stage
Setup: Start from the article’s finite-population-correction mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit variance
Setup: Start from the article’s finite-population-correction mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s finite-population-correction mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s finite-population-correction mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the target population and sampling design, verify the weighting identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.