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eduKateSG · Why Mathematics? · Cochran’s C Statistic, Largest Variances and Laboratory Consistency
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 4–7
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 8–13
Check assumptions
See where design and calibration matter.Route 4 · Chapters 14–18
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 19–22
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when one laboratory appears much more variable than the rest? Cochran's C statistic compares the largest sample variance with the sum of all group variances. The calculation is simple, yet its interpretation depends on replication, model assumptions and a design-specific critical value.
This topic shows a useful habit of mathematical reasoning: first isolate a mechanism, then check whether the assumptions make the number meaningful. A large C is a signal for investigation, not a licence to erase inconvenient data.
Full chapter index
Chapters 1–3 · Understand the question
Chapters 4–7 · Reproduce the example
Chapters 8–13 · Check assumptions
Chapters 14–18 · Explore applications
Chapters 19–22 · Study and verify
Suppose several laboratories apply one procedure to the same material and each produces replicated results. Their within-laboratory variances estimate short-term spread. If one variance dominates, it may point to an execution, instrument, material or recording issue.
Cochran's test is designed for the maximum variance in a set of groups. NIST describes its use in interlaboratory and proficiency contexts, where the identity of an unusually variable laboratory matters.
Let s_i squared be the sample variance in group i. Cochran's original upper-tail statistic is C = largest s_i squared divided by the sum of all s_i squared.
The numerator also appears in the denominator, so C lies between 1/k and 1 when k positive variances are present. If all k variances are equal, C = 1/k. As one variance dominates, C approaches one.
Five laboratories have within-laboratory sample variances 1, 2, 1, 8 and 2 square units. The largest is 8 and the sum is 1 + 2 + 1 + 8 + 2 = 14.
Therefore C = 8/14 = 4/7, approximately 0.5714. Laboratory 4 supplies more than half of the total variance sum in this illustrative set.
| Laboratory | Sample variance | Share of variance sum |
|---|---|---|
| 1 | 1 | 1/14 |
| 2 | 2 | 2/14 |
| 3 | 1 | 1/14 |
| 4 | 8 | 8/14 |
| 5 | 2 | 2/14 |
The critical value depends on the number of groups, the common number of observations per group and the chosen significance level. NIST expresses the upper critical value through an F-distribution quantile. A universal threshold would ignore the design.
The correct next step is to use the governing procedure and its matching critical value. Report k, replication, probability level and whether the test was planned.
| Quantity | Value |
|---|---|
| Largest variance | 8 |
| Sum of variances | 14 |
| C | 8/14 = 4/7 ≈ 0.5714 |
The classical setup assumes the groups have the same number of observations. Unequal replication changes the distribution of sample variances and can invalidate a balanced-design shortcut.
If one laboratory has fewer replicates, its variance is estimated with less precision. Use an appropriate unbalanced method or redesign rather than pretending every variance has the same information.
Cochran's critical-value theory assumes independent normally distributed observations within groups under the null model. Heavy tails or skewness can make a large sample variance more common than the reference distribution expects.
Repeated readings of one unchanged display may also be dependent. Independence comes from how observations are generated, not from separate spreadsheet rows.
The original statistic asks whether the largest variance is unusually large. It does not provide a general test that all variances are identical, and it does not explain why one group is variable.
NIST distinguishes this use from broader homogeneity tests such as Levene or Bartlett procedures. Choose the method from the question.
Mandel's k divides each cell standard deviation by a pooled repeatability standard deviation and supports labelled pattern plots across materials. Cochran's C concentrates on the largest variance share for a material.
The recent article on Mandel's h and k statistics explains the complementary centre-and-spread view.
If results are in milligrams, variance is in square milligrams. The ratio C is dimensionless because squared units cancel. Multiplying every observation by ten multiplies every variance by one hundred but leaves C unchanged.
This scale invariance is a useful arithmetic check. Converting only one laboratory's units would create a false outlier.
Plot every replicate, not just five variances. A single extreme reading can inflate one variance. Alternating high and low results may indicate instability, while a gradual trend may indicate drift.
Check transcription, units, calibration, preparation, timing, environmental conditions and instrument controls. Preserve the original record and any correction history.
An interlaboratory study may test many materials. Applying a separate outlier test to each material creates multiple-testing concerns. The study protocol should state how flags across materials are interpreted.
A laboratory that is variable for every material suggests a different pattern from one that is variable only at a low concentration.
Deleting the largest variance automatically makes the remaining set look more homogeneous. That circular practice can conceal a real limitation of the method.
Exclude only with a documented technical reason and show a sensitivity analysis when useful. A statistical flag begins an investigation.
Five groups can measure the same safe objects with the same written method. Each group should obtain the same number of genuinely repeated results. Calculate each sample variance, their sum and C.
Then inspect the raw values and ask whether any difference reflects resolution, technique, order or object changes. Treat the exercise as process learning, not competition.
Keep one row per raw observation with group, material, replicate, value and unit. Calculate sample variances from the rows. Use MAX for the numerator and SUM for the denominator, while preserving the identity of the maximum group.
Add checks for equal counts, missing values and unit consistency before calculating C.
A program should group data, verify replication, calculate variances with the declared degrees-of-freedom convention, identify ties for the largest variance and compute C. It should not silently fetch a critical value for a different design.
Save the software version, probability level and test direction. Produce a labelled raw-data plot beside the statistic.
C is not the percentage of bad measurements. It is not an accuracy score. A small value does not prove that every laboratory is correct, and a large value does not prove misconduct.
All laboratories can repeat consistently while sharing a common bias. Variance evidence and reference-value evidence must be considered separately.
Students can practise means, squared deviations, sample variance, ratios and probability models. Ask why C cannot exceed one and why equal variances give 1/k.
Parents can encourage students to inspect raw data before judging a group. The most transferable skill is learning to turn a surprising statistic into a fair technical question.
Variance diagnostics appear in analytical chemistry, environmental monitoring, materials testing, manufacturing and calibration. They also teach a broader data skill: preserve group structure and investigate how a summary was generated.
The Mathematics Learning Hub connects this mechanism to school mathematics, while How Mathematics Works develops modelling and assumptions.
Replace the variance 8 with 4 and recompute C. Multiply all raw results by ten and explain why C stays fixed. Add a sixth variance of 2 and calculate the new denominator. Then imagine unequal group sizes and explain why the original critical-value reference may no longer apply.
State the group count, replicate count, unit and next diagnostic in every answer.
### Is Cochran's C the same as Cochran's Q? No. Several procedures carry Cochran's name. The variance-outlier C statistic here is largest variance divided by the sum of variances.
### Can C identify a small variance? The original upper-tail form targets the maximum. Other variants or procedures address different questions.
### Does a significant result justify deletion? No. It justifies technical investigation under the study protocol.
### Can I compare C values from different studies? Only cautiously. Group count, replication, distribution and design affect interpretation.
Use the official NIST Cochran variance outlier test documentation and the NIST h, k and Cochran consistency-plot reference. They document the formula, design-dependent critical values and investigative purpose. The five variances here are original teaching data.
Cochran's C turns five variances into one transparent share. Its usefulness comes from what happens next: verify the design, compare with the correct reference, inspect the raw data and improve the measurement process.
Extended practice laboratory
Practice case 1: Reduce the maximum variance
Setup: Use the article’s Cochran-C example and change one declared input for case 1. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 2: Increase the maximum variance
Setup: Use the article’s Cochran-C example and change one declared input for case 2. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 3: Add an equal group
Setup: Use the article’s Cochran-C example and change one declared input for case 3. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 4: Remove one group
Setup: Use the article’s Cochran-C example and change one declared input for case 4. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 5: Scale every observation
Setup: Use the article’s Cochran-C example and change one declared input for case 5. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 6: Convert only one group incorrectly
Setup: Use the article’s Cochran-C example and change one declared input for case 6. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 7: Create tied maxima
Setup: Use the article’s Cochran-C example and change one declared input for case 7. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 8: Use zero variance
Setup: Use the article’s Cochran-C example and change one declared input for case 8. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 9: Change replication
Setup: Use the article’s Cochran-C example and change one declared input for case 9. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 10: Introduce skewness
Setup: Use the article’s Cochran-C example and change one declared input for case 10. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 11: Insert one extreme value
Setup: Use the article’s Cochran-C example and change one declared input for case 11. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 12: Inspect run order
Setup: Use the article’s Cochran-C example and change one declared input for case 12. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 13: Compare Mandel k
Setup: Use the article’s Cochran-C example and change one declared input for case 13. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 14: Choose a one-sided question
Setup: Use the article’s Cochran-C example and change one declared input for case 14. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 15: Audit sample variance
Setup: Use the article’s Cochran-C example and change one declared input for case 15. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 16: Preserve the flagged group
Setup: Use the article’s Cochran-C example and change one declared input for case 16. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 17: Test a second material
Setup: Use the article’s Cochran-C example and change one declared input for case 17. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 18: Correct a transcription error
Setup: Use the article’s Cochran-C example and change one declared input for case 18. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 19: Check normality
Setup: Use the article’s Cochran-C example and change one declared input for case 19. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 20: Check independence
Setup: Use the article’s Cochran-C example and change one declared input for case 20. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 21: Document alpha
Setup: Use the article’s Cochran-C example and change one declared input for case 21. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 22: Use a design-specific critical value
Setup: Use the article’s Cochran-C example and change one declared input for case 22. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 23: Write a sensitivity analysis
Setup: Use the article’s Cochran-C example and change one declared input for case 23. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 24: State an investigation conclusion
Setup: Use the article’s Cochran-C example and change one declared input for case 24. Task: Recalculate from raw values, preserve units and state which assumption or design condition changed.
Check: Check the relevant invariant, separate the numerical result from the claim it supports, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.