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eduKateSG · Why Mathematics? · Stratified Sampling: Within-Stratum Precision and Weighted Recombination
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 population contains groups that should each be represented and measured precisely? Stratified sampling divides the frame into non-overlapping strata, samples independently inside each, and recombines stratum estimates with population-size weights. Good strata turn known differences into lower sampling variance.
Did You Know? Stratification can improve precision even when every unit has the same overall sampling fraction. The gain comes from replacing one large heterogeneous variance with smaller within-stratum variances.
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
Age bands, regions, school types or business industries can differ greatly. Treating known groups explicitly prevents chance samples from missing important sections.
Strata must be mutually exclusive and collectively exhaustive for the target frame. Every eligible unit belongs to exactly one stratum before selection.
Statistics Canada defines stratification as dividing primary sampling units into homogeneous, mutually exclusive strata and sampling independently in each. Read the official glossary.
Stratum A has N_A=600 and sample mean 70. Stratum B has N_B=400 and sample mean 50.
| Stratum | N | Population weight | Sample mean |
|---|---|---|---|
| A | 600 | 0.6 | 70 |
| B | 400 | 0.4 | 50 |
W_A=600/1000=0.60 and W_B=0.40.
The combined mean is 0.60×70+0.40×50=62.
| Weighted mean | Estimated total |
|---|---|
| 62 | 62000 |
The simple average is 60, which incorrectly gives a 400-unit stratum the same population influence as a 600-unit stratum.
Estimate each total as N_h times its sample mean, then add: 600×70+400×50=62,000. Divide by N=1,000 to recover 62.
Under independent simple random sampling within strata, variance sums stratum contributions N_h²(1/n_h−1/N_h)S_h² for a total.
Statistics Canada gives the stratified HT total and its within-stratum variance formula. Read the official methodology section.
Set n_h proportional to N_h. This produces equal sampling fractions and often self-weighting observations when response and other adjustments do not differ.
Set n_h proportional to N_h S_h under equal costs. More variable strata receive more sample, creating unequal base weights that analysis must use.
Small policy-relevant groups may receive extra sample for reliable domain estimates. Weighting restores their population share in overall estimates.
True stratification controls selection before data collection. Post-stratification adjusts weights after selection to known groups and cannot guarantee sampled units in empty cells.
Sparse or empty achieved samples may force collapsing, but combining unlike strata can reduce precision or alter variance assumptions.
Some large or unique units can form certainty strata. Their sampling variance contribution may be zero at that stage, though measurement and later-stage uncertainty remain.
Strata useful for one outcome may be weak for another. Designers balance key outputs, costs and operational simplicity.
Verify complete partitioning, N_h, n_h, achieved response, within-stratum weights, finite population corrections, lonely PSUs and domain precision.
Reproduce weights 0.60 and 0.40, mean 62 and total 62,000; compare SRS and stratified variance; then oversample B.
Stratification cannot cure a bad frame, within-stratum nonresponse, incorrect N_h, measurement error or hidden heterogeneity.
Decompose a difficult population using reliable prior information, solve within components, then recombine with mathematically correct weights.
Use one row per stratum for N, n, mean, variance, sampling fraction, weight, total contribution and variance contribution.
Validate the frame partition, sample independently by stratum, calculate weights and use survey software with stratum identifiers.
Official statistics, education research, clinical studies and business surveys use stratification.
Build a two-stratum population with different means and variances, compare proportional, Neyman and oversampling designs.
Strata are not analysis groups chosen after outcomes are seen, averaging stratum means equally is usually wrong, and oversampling does not make a group more common.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The Neyman-allocation guide chooses n_h. The finite-population guide handles f_h.
Students should recombine using N_h/N. Parents can ask whether school types were sampled separately and correctly reweighted.
- Calculate two population weights.
- Reproduce mean 62.
- Reproduce total 62,000.
- Oversample B.
- Compare variances.
### Must strata be non-overlapping? Yes.
### Are stratum means equally weighted? Only in special cases.
### Does oversampling bias the result? Not when probabilities and weights are handled correctly.
### Is post-stratification identical? No.
Report target frame, stratum definitions, N_h, n_h, allocation, selection method, weights, response, variance formula, FPCs and collapsed-stratum decisions.
Independent within-stratum estimates become one population estimate only after weighting by the true population shares.
Confirm exhaustive exclusive strata, trusted sizes, independent selection, correct population weights, achieved allocation, stratum-aware variance and transparent domain precision.
Continue through the Mathematics Learning Hub. Recalculate one selection probability and explain which population units the resulting weight represents.
Extended practice laboratory
Practice case 1: Define strata
Setup: Start from the article’s stratified-sampling 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: Check exclusivity
Setup: Start from the article’s stratified-sampling 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: List N values
Setup: Start from the article’s stratified-sampling 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 W values
Setup: Start from the article’s stratified-sampling 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: List means
Setup: Start from the article’s stratified-sampling 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 weighted mean
Setup: Start from the article’s stratified-sampling 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: Calculate totals
Setup: Start from the article’s stratified-sampling 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: Add totals
Setup: Start from the article’s stratified-sampling 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: List variances
Setup: Start from the article’s stratified-sampling 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: Calculate variance contribution
Setup: Start from the article’s stratified-sampling 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: Use proportional allocation
Setup: Start from the article’s stratified-sampling 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: Use Neyman allocation
Setup: Start from the article’s stratified-sampling 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: Oversample small stratum
Setup: Start from the article’s stratified-sampling 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: Calculate weights
Setup: Start from the article’s stratified-sampling 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: Add nonresponse
Setup: Start from the article’s stratified-sampling 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: Add FPC
Setup: Start from the article’s stratified-sampling 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: Collapse strata
Setup: Start from the article’s stratified-sampling 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: Create certainty stratum
Setup: Start from the article’s stratified-sampling 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: Compare post-stratification
Setup: Start from the article’s stratified-sampling 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: Add second outcome
Setup: Start from the article’s stratified-sampling 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 stratified-sampling 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 weights
Setup: Start from the article’s stratified-sampling 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 stratified-sampling 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 stratified-sampling 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 stratified-sampling 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.