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eduKateSG · Why Mathematics? · Mandel’s h and k Statistics for Interlaboratory 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
When several laboratories test the same materials, a table of results can look busy without showing the pattern. Mandel's h and k statistics turn that table into two focused questions: which laboratory means are shifted relative to the group, and which laboratories show unusually large or small within-laboratory spread?
This is why mathematics is important in science and engineering. Standardisation makes unlike-looking rows comparable, while plots keep the laboratory and material labels visible. The statistics are diagnostics, not automatic verdicts.
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 interlaboratory consistency matters
A method may work well in one room and behave differently elsewhere because of calibration, interpretation, environment, preparation or operator practice. An interlaboratory study repeats the same protocol across laboratories and materials to study precision and implementation.
The design must be planned. Laboratories need comparable material, a documented procedure and enough replicates to estimate within-laboratory variation. The numerical pattern is evidence about the study conditions, not a permanent ranking of organisations.
For one material, let each laboratory's cell mean be x-bar_i and let the average of laboratory means be X-bar. Mandel's h is (x-bar_i minus X-bar) divided by the sample standard deviation of the laboratory means. It standardises between-laboratory displacement.
Let s_i be the cell standard deviation within laboratory i. Pool the cell variances as s_r = square root of the mean of s_i squared for a balanced design. Mandel's k is s_i divided by s_r. It standardises within-laboratory spread.
Suppose five laboratories report means 99, 101, 100, 104 and 96 units for one material. Their mean is 100. The deviations are -1, 1, 0, 4 and -4. Their squared deviations sum to 34.
The sample variance of laboratory means is 34/(5 – 1) = 8.5, so their standard deviation is square root of 8.5, about 2.9155 units.
| Lab | Cell mean | Deviation | Within-cell SD |
|---|---|---|---|
| 1 | 99 | -1 | 1.0 |
| 2 | 101 | 1 | 1.5 |
| 3 | 100 | 0 | 0.5 |
| 4 | 104 | 4 | 2.0 |
| 5 | 96 | -4 | 1.0 |
Divide each deviation by 2.9155. The h values are approximately -0.343, 0.343, 0, 1.372 and -1.372. Laboratories 4 and 5 occupy the largest positive and negative positions in this small example.
Their signs show direction. Their magnitudes show displacement relative to the across-laboratory spread. A large positive h means a high laboratory mean for that material; a large negative h means a low one.
Suppose the within-laboratory standard deviations are 1.0, 1.5, 0.5, 2.0 and 1.0 units. Their squared values sum to 8.5, and the balanced pooled repeatability variance is 8.5/5 = 1.7. Thus s_r = square root of 1.7, about 1.3038.
The k values are approximately 0.767, 1.150, 0.383, 1.534 and 0.767. Laboratory 4 has the largest within-laboratory spread; laboratory 3 has the smallest. Neither observation should be converted into a judgement without the study's formal reference lines and records.
| Lab | h | k |
|---|---|---|
| 1 | −0.343 | 0.767 |
| 2 | 0.343 | 1.150 |
| 3 | 0.000 | 0.383 |
| 4 | 1.372 | 1.534 |
| 5 | −1.372 | 0.767 |
A laboratory can have a shifted mean but ordinary repeatability, a centred mean but large repeatability variation, both patterns or neither. h and k separate these mechanisms. That separation points investigators toward calibration and level effects on one hand, or inconsistent execution on the other.
The two plots also work across several materials. Repeated direction across materials can reveal a systematic laboratory pattern that one isolated value would miss.
The simple formulas above assume equal replication in every laboratory-material cell. NIST's current Dataplot documentation notes that ASTM E691 was updated to support unbalanced data while still recommending balanced design. Unbalanced cells require the relevant procedure rather than silent use of the balanced shortcut.
Balance improves comparison because every cell contributes on the same footing. It does not repair non-random missingness or a material that changes during circulation.
Formal h and k consistency plots use critical values that depend on the number of laboratories, replication and chosen probability level. Do not draw a universal line at h = 2 or k = 2 and call it a standard. Use the specified method and version.
NIST describes stringent reference lines used with ASTM E691. Their purpose is to flag patterns for examination. A flag is not permission to discard a laboratory.
Plot materials along one axis and h or k vertically, with a distinct trace for each laboratory. A laboratory that stays positive in h across materials may have a shared calibration or preparation shift. A k spike on one material may signal material-specific handling.
Random sign changes can suggest interaction rather than a simple common offset. Labels matter because follow-up depends on knowing which protocol, instrument and material produced each point.
h compares laboratory means with the consensus of that study. Consensus is not automatically truth. If a traceable assigned value exists, compare results with it separately. A group can agree and still be biased.
k compares spread, not accuracy. A very small k may reflect excellent repeatability, excessive rounding, hidden averaging or too few effective observations. Both tails deserve thought.
The material should be homogeneous and stable enough for the design. Replicates should represent independent executions rather than repeated reading of one prepared result. Units, significant figures and procedures must be aligned.
If laboratories receive materials in a time sequence, transport or ageing may become confounded with laboratory. Randomisation, duplicates and stability checks help protect the interpretation.
Start with raw records: units, transcription, calculations, calibration, preparation and instrument controls. Then inspect whether the unusual result is valid evidence about a method weakness. Removing it only because a plot is uncomfortable defeats the learning purpose.
NIST emphasises that h and k plots are primarily useful for finding systematic patterns and improving the measurement process.
Several groups can measure the mass or length of shared safe objects using the same written procedure. Each group should take independent replicates, preserve raw values and identify its instrument resolution. Calculate cell means, cell standard deviations, h and k.
Discuss whether a shift follows a group, an object or collection order. Avoid turning the exercise into a league table; the aim is to understand sources of variation.
Use one row per raw result with columns for laboratory, material, replicate, value and unit. A pivot table can produce cell counts, means and sample standard deviations. For every material, calculate the mean of cell means and their sample standard deviation.
Pool variances, not standard deviations. Averaging standard deviations gives a different quantity. Add automated checks for equal cell counts before applying balanced formulas.
Group the data by material and laboratory, verify replication, and produce a tidy results table with h and k. Plot labelled values and save the formula version. For an unbalanced design, use a documented implementation that matches the governing procedure.
Recalculate from raw values during review. A polished chart without traceable rows is weak evidence.
CHAPTER 16 OF 22 · Explore applications
16. Connection to recent agreement articles
The Youden plot guide compares two results per laboratory geometrically. Mandel's plots extend the interlaboratory view across materials and replicated cells. The mountain plot guide focuses on a distribution of differences rather than labelled laboratory components.
Choose the graph that preserves the structure your question needs.
Students can practise means, standard deviations, ratios and graphs while learning scientific restraint. Ask them to explain why h has a sign and k cannot be negative. Then ask what extra evidence a flag requires.
Parents can support the habit of keeping raw data and written procedures. The deeper skill is learning to investigate variation without blaming people prematurely.
Interlaboratory reasoning appears in food analysis, environmental monitoring, materials testing, calibration and health laboratories. It also teaches general data skills: standardise carefully, retain group structure, separate centre from spread and investigate patterns.
Explore the broader pathway through the Mathematics Learning Hub and the modelling principles in How Mathematics Works.
Change laboratory 4's mean from 104 to 102 and recompute all h values because the grand mean and standard deviation change. Double every measurement and verify that h and k are unchanged. Replace laboratory 4's within-cell standard deviation 2.0 with 4.0 and recompute s_r and every k.
Explain why changing one cell affects the shared denominator and therefore all laboratories' standardised values.
### Is h a z-score? It has a similar standardising form, but it belongs to a specific interlaboratory consistency procedure with study-dependent reference values.
### Is k a measure of accuracy? No. It compares a laboratory's within-cell standard deviation with a pooled repeatability standard deviation.
### Does a flagged laboratory fail? No. A flag identifies a result or pattern for technical investigation under the applicable protocol.
### Can h and k handle one result per cell? k needs within-cell variation, so replication is essential.
Use the official NIST h- and k-consistency plot documentation, the NIST E691 interlaboratory analysis reference and the NIST metRology software project. They document definitions and purpose; this article's numbers are original teaching data.
Mandel's h and k show mathematics at its most constructive. One ratio reveals relative level, another reveals relative spread, and a labelled plot turns both into questions that teams can answer. The strongest outcome is a better procedure and clearer evidence.
Extended practice laboratory
Practice case 1: Shift one laboratory mean
Setup: Use the article’s Mandel h-and-k 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 every laboratory mean
Setup: Use the article’s Mandel h-and-k 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: Multiply all values
Setup: Use the article’s Mandel h-and-k 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: Increase one cell variance
Setup: Use the article’s Mandel h-and-k 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: Decrease one cell variance
Setup: Use the article’s Mandel h-and-k 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: Add a sixth laboratory
Setup: Use the article’s Mandel h-and-k 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: Remove a flagged cell
Setup: Use the article’s Mandel h-and-k 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: Use unequal replication
Setup: Use the article’s Mandel h-and-k 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: Reverse laboratory labels
Setup: Use the article’s Mandel h-and-k 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: Compare two materials
Setup: Use the article’s Mandel h-and-k 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: Find a persistent positive h
Setup: Use the article’s Mandel h-and-k 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: Find a material-specific k spike
Setup: Use the article’s Mandel h-and-k 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: Check the pooled variance
Setup: Use the article’s Mandel h-and-k 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: Average SDs incorrectly
Setup: Use the article’s Mandel h-and-k 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: Use a traceable value
Setup: Use the article’s Mandel h-and-k 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: Inspect a consensus shift
Setup: Use the article’s Mandel h-and-k 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: Investigate rounding
Setup: Use the article’s Mandel h-and-k 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: Check hidden averaging
Setup: Use the article’s Mandel h-and-k 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: Test material stability
Setup: Use the article’s Mandel h-and-k 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: Plot labelled traces
Setup: Use the article’s Mandel h-and-k 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: Apply design-specific limits
Setup: Use the article’s Mandel h-and-k 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: Preserve all valid data
Setup: Use the article’s Mandel h-and-k 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: Document the protocol version
Setup: Use the article’s Mandel h-and-k 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 a process-improvement conclusion
Setup: Use the article’s Mandel h-and-k 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.