Choose your reading route
eduKateSG · Why Mathematics? · Proficiency-Testing z Scores, Assigned Values and Fitness-for-Purpose Limits
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 in proficiency testing? A participant result can differ from an assigned value by a few units, but that gap is meaningful only relative to the standard deviation chosen for proficiency assessment. The proficiency-testing z score performs exactly that standardisation.
It is a practical lesson in numeracy, statistics and scientific restraint. The formula is short; the hard work lies in choosing a defensible assigned value, a fitness-for-purpose assessment scale and an appropriate action rule.
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
Proficiency testing uses interlaboratory comparisons to evaluate performance for specified measurements and to support continuing improvement. ISO 13528:2022 emphasises that the statistical design must fit the stated purpose.
For a participant result x, assigned value X and standard deviation for proficiency assessment sigma_pt, the score is z = (x – X)/sigma_pt.
Suppose a participant reports x = 103 units. The assigned value is X = 100 units, and the provider has declared sigma_pt = 2 units.
The difference is 103 – 100 = 3 units. Dividing by 2 units gives z = 1.5. The score is dimensionless because the units cancel.
| Participant result x | Assigned value X | sigma_pt | Difference |
|---|---|---|---|
| 103 | 100 | 2 | 3 |
The sign shows direction: positive means the result is above the assigned value, and negative means it is below. The magnitude shows how large the deviation is relative to sigma_pt.
For a symmetric performance rule, +1.5 and -1.5 have the same magnitude but different diagnostic directions. Keep the sign in reports.
| Step | Calculation | Result |
|---|---|---|
| Difference | 103−100 | 3 units |
| Score | 3/2 | z=1.5 |
ISO 13528 notes that a z score above 2 in magnitude is conventionally an indication to investigate possible causes, while a magnitude of 3 or more is conventionally an action signal. Providers define their programme rules and reporting language.
Do not turn a convention into a universal pass mark. The score belongs to one scheme, measurand, round and declared assessment scale.
CHAPTER 5 OF 22 · Reproduce the example
5. The assigned value is not automatically truth
An assigned value can come from a reference measurement, formulation, certified material, expert laboratories or participant consensus. Each route has benefits, uncertainty and possible bias.
The provider should document how X was established. A consensus can be useful without being an exact true value.
The standard deviation for proficiency assessment is the dispersion used to evaluate results. ISO defines it as relating to a hypothetical population performing in accordance with requirements.
It may come from a performance requirement, predictive model, previous precision data or a robust participant distribution. The choice changes every z score, so it must not be hidden.
If x = 104 with the same X = 100 and sigma_pt = 2, z = 2. The result reaches the conventional investigation boundary. If x = 106.5, z = 3.25, exceeding the conventional action-signal magnitude.
These calculations show why rounding must follow the scheme rule. Calculate from unrounded inputs before displaying a score.
The basic z score does not include the participant's stated measurement uncertainty in its denominator. It also does not include assigned-value uncertainty directly.
Other scores address those questions. A z-prime score includes assigned-value uncertainty, while a zeta score combines participant and assigned-value standard uncertainties.
ISO 13528 explains advantages of choosing assigned values and assessment criteria independently of participant results, especially for a consistent fitness-for-purpose criterion. Participant-derived values can make the group influence its own evaluation.
When consensus is used, robust methods and uncertainty assessment become important.
A score can reflect a transcription error, unit conversion, preparation problem, instrument drift, sample instability or chance. Investigate the measurement process and the raw record.
Long-term trends can show repeated direction, but rounds may differ in material, assigned value and assessment scale. Compare only after checking comparability.
Plot signed scores by analyte, concentration or round. Add the scheme's warning and action lines. Label results so records can be traced.
A histogram of participant scores can show centre and spread, but it does not replace the provider's assigned-value method.
z scores were designed for performance evaluation against a criterion, not simplistic league tables. Differences between 0.2 and 0.4 may not represent meaningful capability differences.
Focus on investigation, corrective action and learning. Preserve uncertainty and design information.
Several groups can measure a safe common object. The teacher provides a documented assigned value and a declared classroom assessment scale. Students calculate signed deviations and z scores.
The activity is an analogy, not accredited proficiency testing. Its purpose is to connect subtraction, division, units and evidence.
Create columns for participant result, assigned value, sigma_pt, difference, z, absolute z and signal category. Validate that sigma_pt is positive and units match.
Keep the source of X and sigma_pt in metadata. Conditional colours should follow, not define, the calculation.
A program should validate inputs, compute signed scores with full precision and apply the declared inclusive or exclusive boundaries. It should preserve missing and invalid results rather than silently dropping them.
Test x = 103, X = 100 and sigma_pt = 2; the result must be 1.5.
CHAPTER 16 OF 22 · Explore applications
16. Connection to recent mathematics articles
The HorRat article explains a concentration-dependent precision benchmark. The normalised-error article scales a difference by combined expanded uncertainty. A PT z score uses sigma_pt instead.
Students can ask three questions: what is the assigned value, where did sigma_pt come from, and what action rule applies? A good answer includes arithmetic and a limited conclusion.
Parents can encourage investigation rather than shame. A surprising score is a prompt to improve a process.
Proficiency scores appear in analytical chemistry, calibration, environmental testing, food laboratories and health-related measurement programmes. Mathematics supports comparability and quality systems, while professional protocols define consequences.
Explore foundations through the Mathematics Learning Hub and How Mathematics Works.
Calculate z for x = 97, X = 100 and sigma_pt = 2. Double all three unit-valued quantities and verify the score. Halve sigma_pt and explain why the magnitude doubles. Change X by one unit and track the sign.
For each answer, state the scheme scale and the limited action it supports.
### Is this the same as a classroom z score? The algebra is similar, but PT uses an assigned value and provider-chosen standard deviation for proficiency assessment rather than automatically using the sample mean and sample SD.
### Does |z| below 2 prove accuracy? No. It indicates consistency with the scheme's criterion for that result.
### Should participant uncertainty be ignored? No. It can be assessed separately or through a score such as zeta, depending on the scheme.
### Can sigma_pt be chosen after seeing results? That can create circular evaluation. The provider should document a defensible method aligned with the scheme purpose.
Use the official ISO 13528:2022 online standard page, the NIST ISO 13528 z-score documentation, the Eurachem PT guide and the NIST Office of Weights and Measures PT statistical procedure. The worked values here are original.
A proficiency-testing z score turns a difference into a scale-aware question. Its strength comes from transparent value assignment, a purposeful assessment standard deviation and a fair investigation after the number appears.
Extended practice laboratory
Practice case 1: Move the result upward
Setup: Use the article’s PT-z-score example and change one declared input for case 1. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Move the result downward
Setup: Use the article’s PT-z-score example and change one declared input for case 2. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Double all unit values
Setup: Use the article’s PT-z-score example and change one declared input for case 3. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Halve sigma_pt
Setup: Use the article’s PT-z-score example and change one declared input for case 4. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Change the assigned value
Setup: Use the article’s PT-z-score example and change one declared input for case 5. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Reach magnitude two
Setup: Use the article’s PT-z-score example and change one declared input for case 6. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Cross magnitude three
Setup: Use the article’s PT-z-score example and change one declared input for case 7. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Reverse the sign
Setup: Use the article’s PT-z-score example and change one declared input for case 8. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Round near a boundary
Setup: Use the article’s PT-z-score example and change one declared input for case 9. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Use a consensus X
Setup: Use the article’s PT-z-score example and change one declared input for case 10. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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 an independent X
Setup: Use the article’s PT-z-score example and change one declared input for case 11. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Change the fitness criterion
Setup: Use the article’s PT-z-score example and change one declared input for case 12. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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 two analytes
Setup: Use the article’s PT-z-score example and change one declared input for case 13. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Plot round-to-round signs
Setup: Use the article’s PT-z-score example and change one declared input for case 14. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Check unit conversion
Setup: Use the article’s PT-z-score example and change one declared input for case 15. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Handle a missing result
Setup: Use the article’s PT-z-score example and change one declared input for case 16. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Audit sigma_pt origin
Setup: Use the article’s PT-z-score example and change one declared input for case 17. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Audit X uncertainty
Setup: Use the article’s PT-z-score example and change one declared input for case 18. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Compare z-prime
Setup: Use the article’s PT-z-score example and change one declared input for case 19. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Compare zeta
Setup: Use the article’s PT-z-score example and change one declared input for case 20. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Avoid ranking
Setup: Use the article’s PT-z-score example and change one declared input for case 21. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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 a transcription error
Setup: Use the article’s PT-z-score example and change one declared input for case 22. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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: Record the scheme rule
Setup: Use the article’s PT-z-score example and change one declared input for case 23. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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 result-limited conclusion
Setup: Use the article’s PT-z-score example and change one declared input for case 24. Task: Recalculate from raw values, preserve units and state which scoring or design assumption changed.
Check: Check the relevant invariant, separate the number from the claim it supports, and name the investigation or source needed next. 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.