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eduKateSG · Why Mathematics? · Repeatability Coefficients, Within-Subject Variation and Expected Pair Differences
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 two measurements of the same thing do not match exactly? The repeatability coefficient turns that everyday puzzle into a useful boundary. It estimates how far apart two results may be under stated repeatability conditions, so a student can distinguish ordinary short-term variation from a difference that deserves investigation.
This is one concrete benefit of learning mathematics: a formula becomes a disciplined way to compare evidence. The coefficient does not certify that an instrument is correct, and it does not promise that every future pair will lie inside the boundary. It describes variation under a design, with assumptions and uncertainty that must be reported.
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
CHAPTER 1 OF 22 · Understand the question
1. Why repeatability is a mathematical question
Repeatability asks what happens when the same measurand is measured again under conditions kept as similar as practical. NIST describes repeatability conditions through the same procedure, observer, instrument, location and a short time interval. The wording matters because changing an operator, day, laboratory or method introduces other sources of variation.
Mathematics helps by separating a visible difference between two readings from the underlying spread of repeated results. A single close pair is encouraging but weak evidence. Many planned pairs allow a standard deviation, a bias check and a repeatability coefficient to be calculated.
CHAPTER 2 OF 22 · Understand the question
2. The mechanism: differences, standard deviations and a boundary
For paired measurements, define each difference as d_i = result 1 minus result 2. Let s_d be the sample standard deviation of those differences. When the two repeats have similar variance and independent short-term errors, the within-subject standard deviation can be estimated as s_w = s_d divided by square root of two.
A widely used 95% repeatability coefficient is 1.96 times square root of two times s_w. With paired differences this simplifies to 1.96s_d. Under an approximately normal, stable difference model, about 95% of differences around their mean are expected within mean difference plus or minus this coefficient. The mean difference must be shown separately if it is not negligible.
Suppose a student measures six calibration objects twice in the same session, using the same instrument and procedure. The paired differences are 1, -1, 2, -2, 0 and 2 units. Their sum is 2, so the mean difference is 2/6 = 1/3 unit.
The deviations from 1/3 are 2/3, -4/3, 5/3, -7/3, -1/3 and 5/3. Their squared deviations add to 120/9 = 40/3. Dividing by n – 1 = 5 gives a sample variance of 8/3, so s_d = square root of 8/3, about 1.633 units.
| Item | Difference d | d − 1/3 | Squared deviation |
|---|---|---|---|
| 1 | 1 | 2/3 | 4/9 |
| 2 | -1 | −4/3 | 16/9 |
| 3 | 2 | 5/3 | 25/9 |
| 4 | -2 | −7/3 | 49/9 |
| 5 | 0 | −1/3 | 1/9 |
| 6 | 2 | 5/3 | 25/9 |
The within-subject variance estimate is s_d squared divided by two: (8/3)/2 = 4/3. Therefore s_w = square root of 4/3, about 1.155 units. The repeatability coefficient is 1.96 × square root of two × 1.155, which equals 1.96 × 1.633, about 3.201 units.
The fitted 95% difference interval is centred on the observed mean difference: 0.333 ± 3.201, or approximately -2.868 to 3.534 units. Reporting only ±3.201 would hide the estimated direction of the shift.
| Quantity | Exact | Approximate |
|---|---|---|
| Mean difference | 1/3 | 0.333 |
| s_d² | 8/3 | 2.667 |
| s_d | √(8/3) | 1.633 |
| s_w | √(4/3) | 1.155 |
| Repeatability coefficient | 1.96√(8/3) | 3.201 |
The coefficient has the same unit as the measurement. A value of 3.201 units means that paired results under the represented conditions may commonly differ by roughly that amount in either direction around the mean difference. Whether that is acceptable depends on a tolerance chosen from the purpose of the measurement.
It does not prove accuracy. Two readings can repeat beautifully while both are displaced from a traceable reference. It also does not describe changed operators or laboratories unless the design deliberately included them.
CHAPTER 6 OF 22 · Reproduce the example
6. Repeatability, reproducibility and stability
Repeatability keeps conditions close. Reproducibility deliberately changes stated conditions, such as the observer, instrument, location or time. Stability asks whether the process stays consistent across time. These questions require different designs and usually different variance components.
NIST cautions that day-to-day measurement variation can exceed short-term precision. A small repeatability coefficient can therefore coexist with drift, environmental sensitivity or operator effects. Name the conditions before naming the statistic.
Let the two independent repeat errors each have variance sigma squared. The variance of their difference is sigma squared plus sigma squared, because variances add for independent errors. The difference therefore has standard deviation square root of two times sigma.
This reasoning is more valuable than memorising the formula. It shows why correlated errors change the result: if repeats share a common disturbance, covariance enters the variance of the difference. The coefficient inherits the design's dependence structure.
The mean difference in the example is 1/3 unit. That small sample estimate may reflect a systematic order effect, warm-up drift or chance. Plot differences in collection order and against the average of each pair. Look for a non-zero centre, trends and larger spread at larger magnitudes.
Precision and bias answer different questions. A narrow distribution centred away from zero is repeatable but shifted. A wide distribution centred near zero is unbiased on average but poorly repeatable.
The familiar multiplier 1.96 comes from a normal distribution and an estimated stable spread. With only six pairs, tail estimation and normality checks are weak. The coefficient is a point estimate, not a guarantee, and its own sampling uncertainty can be substantial.
Independence, representative sampling, consistent units, a meaningful pairing rule and a stable protocol all matter. If repeated observations are clustered within a few objects or students, ordinary formulas can overstate the effective sample size.
Choose objects that span the intended measurement range. Randomise or counterbalance order when order effects are plausible. Record environmental conditions, operator, instrument identifier, timing, resolution and any failed reading. Predefine how missing and rounded values will be handled.
More objects usually teach more about transfer than many rapid repeats of one object. Replicates within each object help estimate short-term spread, while range coverage reveals whether variability grows with magnitude.
When spread grows with the measurement level, a constant-unit coefficient may be misleading. A log transformation can turn multiplicative error into an additive difference. Back-transformed limits then become ratios rather than fixed-unit gaps.
Percentage differences can be useful, but denominators near zero are unstable. State the chosen reference in every relative calculation. Do not switch between percentage points and percentages.
CHAPTER 12 OF 22 · Check assumptions
12. A plot can expose what one coefficient hides
A differences-against-means plot shows whether spread changes across the range. A run-order plot can reveal warming or drift. A histogram or quantile plot can expose skewness and outliers. Label unusual observations so the raw record can be checked.
The recently published guide to Bland–Altman limits and paired differences develops this visual logic. The guide to coverage probability asks a complementary fixed-tolerance question.
Rounding can create false ties. Selective deletion can shrink the estimated spread. A changing measurand can be mistaken for instrument noise. Reusing the same specimen may introduce memory, fatigue or depletion effects. A convenient classroom sample may not represent field conditions.
Investigate an extreme observation before deciding whether it is an error. Keep both the analysis with all valid data and any justified sensitivity analysis.
Measure the same set of objects twice with a ruler, balance or timer. Use a safe, non-sensitive task. Create columns for object, first result, second result, difference, pair mean and collection order. Calculate the mean difference, s_d, s_w and repeatability coefficient.
Then change one condition deliberately, such as the observer or day. Compare the new spread with the original repeatability condition. Describe the design change instead of saying one instrument is simply good or bad.
Keep raw measurements in separate columns. Compute each difference with a transparent formula. Use the sample standard deviation function, not the population version, when the observations are a sample from future conditions. Calculate the coefficient in a labelled cell and preserve full precision until display.
Add checks for count, missing values and units. A result that cannot be traced to rows is difficult to audit.
A short program should read immutable raw data, validate pairing, compute differences, report the mean and sample standard deviation, and produce diagnostic plots. Record the software and formula convention. If resampling confidence intervals are used, resample the independent object, not individual rows inside a cluster.
Code does not replace design judgement. It makes the declared arithmetic repeatable.
Students can learn this topic through arithmetic, algebra, graphs and probability. Ask what stayed the same, what changed, and what the unit means. A correct answer should include a calculation and a sentence limiting the claim.
Parents can encourage careful recording rather than a race for a tiny coefficient. A surprising result is often the best prompt to inspect method, order, resolution and context.
CHAPTER 18 OF 22 · Explore applications
18. Mathematics in laboratories, engineering and health research
Repeatability reasoning appears in calibration, manufacturing, environmental monitoring, sports measurement and clinical method studies. The mathematics supports decisions, but subject experts define tolerances and consequences. Career pathways remain open: statistics, science, engineering, quality and data work all use the same habit of connecting a number to a design.
The Mathematics Learning Hub connects this mechanism to school mathematics, while How Mathematics Works develops the wider ideas of modelling and proof.
Recalculate the example after multiplying every reading by ten. Then add a constant two units to both measurements and explain what changes. Replace one difference of 2 with 8 and compare the mean difference and coefficient. Finally, decide whether a tolerance of ±2 units is compatible with the fitted interval.
Each answer should state the unit, the condition represented and one limitation.
### Is the repeatability coefficient the same as correlation? No. Correlation measures association and can remain high when paired values disagree by a fixed amount. Repeatability focuses on the spread of differences under stated conditions.
### Does 95% mean exactly 95 of the next 100 pairs? No. It is a model-based long-run statement with estimated parameters. A particular future set can differ, especially when the process or population changes.
### Should a coefficient always be as small as possible? Smaller is better only relative to a declared measurement purpose. The study must also cover realistic objects and conditions.
### Can six pairs establish repeatability? They can teach the arithmetic, but they provide limited evidence. A real study needs a justified sample size, representative range and uncertainty assessment.
Use the official NIST definition of repeatability, the NIST repeatability standard deviation reference and NIST terminology for repeatability and reproducibility. These sources define conditions and spread; the worked numbers here are original teaching data.
Repeatability is a cheerful example of mathematics doing careful work. Subtraction exposes paired variation, variance explains why errors combine, and probability supplies a boundary. The best conclusion is modest and useful: under clearly stated conditions, this is the variation the data support, and this is what should be checked next.
Extended practice laboratory
Practice case 1: Scale every value
Setup: Use the article’s paired-repeatability example and change one declared input for case 1. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Shift both repeats
Setup: Use the article’s paired-repeatability example and change one declared input for case 2. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Reverse the subtraction
Setup: Use the article’s paired-repeatability example and change one declared input for case 3. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Insert one extreme difference
Setup: Use the article’s paired-repeatability example and change one declared input for case 4. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Remove a failed pair
Setup: Use the article’s paired-repeatability example and change one declared input for case 5. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Double the pair count
Setup: Use the article’s paired-repeatability example and change one declared input for case 6. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Cluster repeats by object
Setup: Use the article’s paired-repeatability example and change one declared input for case 7. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Change the observer
Setup: Use the article’s paired-repeatability example and change one declared input for case 8. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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 the day
Setup: Use the article’s paired-repeatability example and change one declared input for case 9. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Add rounding
Setup: Use the article’s paired-repeatability example and change one declared input for case 10. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Use a log scale
Setup: Use the article’s paired-repeatability example and change one declared input for case 11. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Use percentage differences
Setup: Use the article’s paired-repeatability example and change one declared input for case 12. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Test range dependence
Setup: Use the article’s paired-repeatability example and change one declared input for case 13. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Check collection order
Setup: Use the article’s paired-repeatability example and change one declared input for case 14. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Compare a fixed tolerance
Setup: Use the article’s paired-repeatability example and change one declared input for case 15. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Estimate a confidence interval
Setup: Use the article’s paired-repeatability example and change one declared input for case 16. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Plot differences against means
Setup: Use the article’s paired-repeatability example and change one declared input for case 17. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Separate bias from spread
Setup: Use the article’s paired-repeatability example and change one declared input for case 18. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Audit independence
Setup: Use the article’s paired-repeatability example and change one declared input for case 19. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Define the measurand
Setup: Use the article’s paired-repeatability example and change one declared input for case 20. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Record instrument resolution
Setup: Use the article’s paired-repeatability example and change one declared input for case 21. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Investigate an outlier
Setup: Use the article’s paired-repeatability example and change one declared input for case 22. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Plan representative sampling
Setup: Use the article’s paired-repeatability example and change one declared input for case 23. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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: Write an evidence-limited conclusion
Setup: Use the article’s paired-repeatability example and change one declared input for case 24. Task: Recalculate from the raw values, preserve units and state which design condition changed.
Check: Check arithmetic against the invariant described in the chapter, then separate the numerical finding from the practical claim. 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.