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eduKateSG · Why Mathematics? · z′ Scores, Assigned-Value Uncertainty and Proficiency Assessment
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 19–23
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
What happens when the assigned value in a proficiency test is itself uncertain? The z-prime score, written z′, adds the assigned value's standard uncertainty to the proficiency-assessment variance before standardising the participant's deviation.
This is why mathematics matters: uncertainty is not an afterthought. It changes the denominator and therefore changes how strongly a difference should be interpreted.
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 19–23 · Study and verify
For participant result x, assigned value X, standard deviation for proficiency assessment sigma_pt and standard uncertainty u(X), z′ = (x – X)/square root of[sigma_pt squared + u(X) squared].
NIST's ISO 13528 implementation uses these quantities as provider-defined inputs because several valid routes exist for determining them.
Suppose x = 104 units, X = 100 units, sigma_pt = 2 units and u(X) = 1.5 units. The signed difference is 4 units.
The denominator is square root of 2 squared plus 1.5 squared = square root of 4 + 2.25 = square root of 6.25 = 2.5 units. Therefore z′ = 4/2.5 = 1.6.
| x | X | sigma_pt | u(X) |
|---|---|---|---|
| 104 | 100 | 2 | 1.5 |
Ignoring assigned-value uncertainty, z would be 4/2 = 2. Including u(X) gives z′ = 1.6. The assigned value is not treated as perfectly known, so the standardised magnitude is smaller.
This is not a reward or discount. It is a different model of the comparison evidence.
| Step | Calculation | Result |
|---|---|---|
| Difference | 104−100 | 4 |
| Denominator | √(2²+1.5²) | 2.5 |
| z-prime | 4/2.5 | 1.6 |
Under independence, the uncertainty variance of a difference includes the variance of the assigned value. Sigma_pt squared and u(X) squared are both in squared units, so they add before taking the square root.
If dependence exists, covariance may need to be included under the scheme's method. Root-sum-of-squares is not a substitute for checking shared inputs.
If u(X) is small compared with sigma_pt, z′ is close to z. If it is substantial, the difference can be material. Some programmes use a stated ratio such as u(X)/sigma_pt to decide whether ordinary z scoring is adequate.
Follow the current provider procedure instead of inventing a universal cutoff.
With the same x, X and sigma_pt but u(X) = 0.5, the denominator is square root of 4.25, about 2.0616. z′ is about 1.9403.
As u(X) approaches zero, z′ approaches z. This limiting case is an excellent formula check.
CHAPTER 7 OF 23 · Reproduce the example
7. Keep standard and expanded uncertainty distinct
The formula uses the standard uncertainty of the assigned value, not an expanded uncertainty unless the provider's notation explicitly converts it. If U = ku, divide by the documented coverage factor before substituting u.
Mixing coverage conventions can distort the score.
A large u(X) can reduce |z′|, but it also means the reference is less precise. It should prompt attention to value assignment, homogeneity, stability and reference methods.
The goal is not to manufacture comfortable scores. The goal is an honest evaluation.
NIST's ISO 13528 z-prime documentation describes conventional warning and action magnitudes at 2 and 3. The provider defines how boundaries are applied and what investigation follows.
Report the signed score, inputs and rule. Avoid rounding a borderline result before classification.
z′ combines sigma_pt with assigned-value uncertainty. Zeta combines the participant result's standard uncertainty with the assigned value's standard uncertainty.
One asks about performance against a proficiency criterion; the other checks consistency with stated uncertainties.
Normalised error commonly uses expanded uncertainties in a measurement-comparison context. z′ uses sigma_pt and the standard uncertainty of X.
The formulas resemble one another because both standardise a difference, but the denominators represent different purposes.
If the participant result helped determine X, or both depend on a shared standard, independence can fail. The provider should account for or limit such dependence.
An assigned value independent of participant results often simplifies interpretation.
Each analyte can have a different X, sigma_pt and u(X). Plot scores with those inputs available. A run of positive scores may indicate bias, while mixed isolated scores may suggest local issues.
Do not average z′ values blindly across incomparable measurands.
Start with one difference and fixed sigma_pt. Vary u(X) from zero upward and graph z′. The curve shows that greater reference uncertainty lowers the standardised magnitude.
Then explain why that does not improve the measurement itself.
Use separate cells for x, X, sigma_pt and u(X). Calculate difference, combined denominator, signed z′ and absolute z′. Validate positivity and units.
Keep any conversion from expanded to standard uncertainty visible.
Reject negative uncertainties and zero denominators. Compute with unrounded inputs, preserve the sign and attach the scheme's rule version. Add unit tests for u(X)=0 and the 3-4-5 triangle example with sigma_pt=2, u(X)=1.5.
The expected denominator is exactly 2.5.
Students can connect Pythagorean geometry, variance, units and decision thresholds. Ask why the denominator grows but the numerator does not.
Parents can encourage the language “less precisely assigned” instead of “more correct.”
z-prime reasoning appears in laboratory proficiency testing and quality systems where assigned values have non-negligible uncertainty. It teaches a general skill: propagate uncertainty before judging a difference.
The Mathematics Learning Hub and How Mathematics Works develop the school and modelling foundations.
The ordinary proficiency z score explains the base criterion without assigned-value uncertainty. The degrees-of-equivalence article retains a difference with its uncertainty rather than reducing it to this score.
Calculate z′ for x = 97, X = 100, sigma_pt = 2 and u(X) = 1.5. Set u(X) to zero and compare. Double every unit-valued input and verify invariance. Convert an expanded uncertainty U = 3 with k = 2 into standard uncertainty before scoring.
Explain which change affects the numerator and which affects the denominator.
### Why is there a prime mark? It distinguishes this variation from the ordinary PT z score by including assigned-value uncertainty.
### Does a larger u(X) make performance better? No. It makes the comparison reference less precise and changes the evaluation model.
### Can I substitute participant uncertainty for sigma_pt? No. That creates a different score, usually zeta.
### Is z′ dimensionless? Yes. Numerator and denominator have the same unit.
Use the official ISO 13528:2022 definitions, the NIST ISO 13528 z-prime documentation, the Eurachem PT guide and the IAEA proficiency-testing data-analysis example. The worked numbers are original.
z-prime makes one modest but important correction: the assigned value has uncertainty too. By putting that uncertainty into the denominator, mathematics helps a proficiency decision match the evidence actually available.
Extended practice laboratory
Practice case 1: Set assigned uncertainty to zero
Setup: Use the article’s z-prime 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: Increase assigned uncertainty
Setup: Use the article’s z-prime 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: Decrease assigned uncertainty
Setup: Use the article’s z-prime 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: Double all unit values
Setup: Use the article’s z-prime 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: Reverse the subtraction
Setup: Use the article’s z-prime 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 a 3-4-5 denominator
Setup: Use the article’s z-prime 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: Convert expanded uncertainty
Setup: Use the article’s z-prime 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: Mix uncertainty conventions incorrectly
Setup: Use the article’s z-prime 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: Add correlation
Setup: Use the article’s z-prime 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: Compare ordinary z
Setup: Use the article’s z-prime 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: Compare zeta
Setup: Use the article’s z-prime 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: Compare En
Setup: Use the article’s z-prime 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: Change sigma_pt
Setup: Use the article’s z-prime 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: Change X
Setup: Use the article’s z-prime 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: Round near two
Setup: Use the article’s z-prime 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: Round near three
Setup: Use the article’s z-prime 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 value assignment
Setup: Use the article’s z-prime 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 homogeneity
Setup: Use the article’s z-prime 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: Audit stability
Setup: Use the article’s z-prime 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: Plot several analytes
Setup: Use the article’s z-prime 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: Preserve the sign
Setup: Use the article’s z-prime 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: Document covariance
Setup: Use the article’s z-prime 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: Document scheme rules
Setup: Use the article’s z-prime 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 an uncertainty-aware conclusion
Setup: Use the article’s z-prime 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.