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eduKateSG · Why Mathematics? · Rescaled Sums of z Scores, Persistent Bias and Multianalyte Patterns
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
One proficiency score can look harmless while a laboratory shows a small bias in the same direction across many analytes. The rescaled sum of z scores, RSZ, adds signed z scores and divides by the square root of their count. It is designed to reveal coherent directional patterns.
This is why mathematics matters in quality systems: weak signals can accumulate. The method works only when the scores are suitably comparable and their dependence is understood.
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
Let z_1 through z_L be proficiency z scores for L results. RSZ = sum(z_i)/square root of L.
If scores are independent standard normal variables under a suitable null model, their sum has variance L, so division by square root of L returns unit variance.
Suppose four analyte scores are 1.2, 0.8, 1.5 and 0.5. Their sum is 4.0 and square root of 4 is 2.
RSZ = 4/2 = 2.0. Every score is positive, so the combined statistic reveals a consistent upward tendency.
| Score 1 | Score 2 | Score 3 | Score 4 | Sum |
|---|---|---|---|---|
| 1.2 | 0.8 | 1.5 | 0.5 | 4.0 |
The mean score would be 1.0, but its standard deviation under independence is 1/square root of L. Multiplying the mean by square root of L gives RSZ.
Dividing the sum by L answers a different descriptive question.
| Quantity | Calculation | Result |
|---|---|---|
| L | Count scores | 4 |
| sqrt(L) | sqrt(4) | 2 |
| RSZ | 4/2 | 2.0 |
Scores 1.2, -0.8, 1.5 and -0.5 sum to 1.4, giving RSZ = 0.7. Large mixed-direction errors can cancel.
Use the sum of squared z scores or individual plots to detect non-directional excess scatter. RSZ is intentionally sensitive to consistent bias.
Analytes from one sample may share preparation, calibration, blank or instrument effects. Positive correlation makes the variance of the sum exceed L, so the simple RSZ can look more extreme than the independence model predicts.
Use a covariance-aware statistic or a documented scheme procedure when scores are correlated.
Each z score should have a meaningful assigned value and sigma_pt. Combining scores from unrelated schemes, rounds or purposes can create a number without a coherent interpretation.
Predefine which analytes enter the sum and how missing results are handled.
Choosing only positive scores inflates RSZ. Selecting analytes after inspection destroys the reference distribution.
The set must be declared independently of the observed signs.
If one analyte is missing, L changes. A missing result caused by measurement failure may be informative and should not disappear without documentation.
Report included analytes, exclusions and the reason for each.
For z scores -1.7, -0.9, -1.2 and -0.6, the sum is -4.4 and RSZ = -2.2. The negative sign indicates a coherent low tendency.
Investigate shared calibration or preparation effects rather than treating four scores as unrelated.
RSZ can combine analytes within a round or comparable scores across time, but those designs have different dependence structures. Round-to-round material changes can break comparability.
State the axis of aggregation.
Plot RSZ by round to monitor directional changes. Control limits require a justified dependence and stability model.
A chart supports investigation; it does not prove a cause.
The sum of squared z scores, SSZ = sum(z_i squared), ignores direction and reacts to large magnitudes. Under independence it can be compared with a chi-square reference with L degrees of freedom.
RSZ and SSZ answer complementary centre and spread questions.
Give students sets of four signed scores. Compare individual magnitudes, RSZ and SSZ. Create one set with consistent small bias and one with alternating large signs.
Ask which statistic detects which pattern.
Use one row per declared score with analyte, round, sign, inclusion status and reason. Sum included scores, count L and divide by square root of L.
Calculate SSZ in a separate column and never hide the individual scores.
Validate the prespecified set, handle missingness explicitly and calculate both RSZ and score covariance when available. Test four scores summing to four to obtain RSZ=2.
Return inclusion metadata with the number.
RSZ is not an average proficiency grade. A near-zero value can hide large opposing scores. A large value does not identify the shared cause.
Independence is an assumption, not a default truth.
Students can connect sums, variance addition, square roots and correlation. Ask why four independent unit-variance scores have a sum SD of two.
Parents can encourage inspection of the component scores. Aggregates should invite questions, not replace evidence.
Combined scores can support laboratory trend monitoring, sampling proficiency and multianalyte quality review. The transferable skill is aggregating weak signals while respecting dependence.
Use the Mathematics Learning Hub and How Mathematics Works for broader learning.
The Birge-ratio guide checks overall uncertainty consistency. The PT z-score guide defines each component score.
Calculate RSZ for 1,1,1,1 and for 2,-2,2,-2. Compute SSZ for both sets. Add a fifth zero score and explain how the denominator changes. Assume pairwise correlation 0.25 and explain why square root of L is no longer the correct SD of the sum.
### Why divide by square root of L? Under independence, variances add to L, so the sum's standard deviation is square root of L.
### Can opposite errors cancel? Yes. That is why individual scores and SSZ remain important.
### Can I combine any z scores? No. They need a coherent design, comparable meaning and declared dependence assumptions.
### Is RSZ a formal pass/fail score? Only if the governing programme defines and validates such a rule.
Use the Eurachem sampling-uncertainty workshop example, which shows RSZ as sum z divided by square root of n, the IAEA XRF comparison discussion and the Eurachem PT guide. The worked scores are original.
RSZ shows how several modest signed signals can reveal a shared direction. Mathematics makes the accumulation fair under stated assumptions and reminds us to inspect correlation, selection and every component score.
Extended practice laboratory
Practice case 1: Sum four positive scores
Setup: Use the article’s RSZ example and change one declared input for case 1. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 2: Sum alternating signs
Setup: Use the article’s RSZ example and change one declared input for case 2. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 3: Compute SSZ
Setup: Use the article’s RSZ example and change one declared input for case 3. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 4: Add a zero score
Setup: Use the article’s RSZ example and change one declared input for case 4. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 5: Add a fifth positive score
Setup: Use the article’s RSZ example and change one declared input for case 5. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 6: Reverse all signs
Setup: Use the article’s RSZ example and change one declared input for case 6. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 7: Average instead of rescale
Setup: Use the article’s RSZ example and change one declared input for case 7. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 8: Check variance of the sum
Setup: Use the article’s RSZ example and change one declared input for case 8. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 9: Introduce positive correlation
Setup: Use the article’s RSZ example and change one declared input for case 9. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 10: Introduce negative correlation
Setup: Use the article’s RSZ example and change one declared input for case 10. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 11: Select only positives
Setup: Use the article’s RSZ example and change one declared input for case 11. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 12: Prespecify analytes
Setup: Use the article’s RSZ example and change one declared input for case 12. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 13: Handle missing data
Setup: Use the article’s RSZ example and change one declared input for case 13. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 14: Combine two rounds
Setup: Use the article’s RSZ example and change one declared input for case 14. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 15: Plot RSZ by round
Setup: Use the article’s RSZ example and change one declared input for case 15. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 16: Inspect component scores
Setup: Use the article’s RSZ example and change one declared input for case 16. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 17: Find cancellation
Setup: Use the article’s RSZ example and change one declared input for case 17. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 18: Find persistent bias
Setup: Use the article’s RSZ example and change one declared input for case 18. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 19: Compare Birge ratio
Setup: Use the article’s RSZ example and change one declared input for case 19. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 20: Compare a control chart
Setup: Use the article’s RSZ example and change one declared input for case 20. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 21: Check comparable sigma_pt
Setup: Use the article’s RSZ example and change one declared input for case 21. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 22: Audit inclusion metadata
Setup: Use the article’s RSZ example and change one declared input for case 22. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 23: Avoid pass-fail overclaim
Setup: Use the article’s RSZ example and change one declared input for case 23. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 24: Write a dependence-limited conclusion
Setup: Use the article’s RSZ example and change one declared input for case 24. Task: Recalculate from raw values, preserve units and state which scoring or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.