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eduKateSG · Why Mathematics? · Sum of Squared z Scores, Error Magnitude and Multianalyte Performance
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 19–24
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a laboratory reports many analytes at once? A single acceptable-looking summary can hide large errors with opposite signs. The sum of squared z scores, or SSZ, removes signs by squaring each score and then adds the magnitudes into one multianalyte diagnostic.
SSZ connects algebra, vectors, chi-squared reasoning and quality improvement. It is especially useful beside the rescaled sum of z scores, because the two summaries see different patterns.
Did you know? A positive error and an equally large negative error cancel in an ordinary sum, but their squares reinforce each other.
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 19–24 · Study and verify
Suppose a participant reports results for several analytes, each with a valid assigned value and standard deviation for proficiency assessment. Each result can be converted to a z score.
SSZ asks how much total squared standardized departure appears across the declared set. It ignores direction so cancellation cannot hide magnitude.
For L included z scores, the IAEA defines SSZ = sum from i = 1 to L of z_i squared. Every included score contributes a non-negative amount.
The inclusion set matters. The IAEA example sums analytes with known assigned values reported by the participant.
| z1 | z2 | z3 | z4 |
|---|---|---|---|
| 1.2 | -1.1 | 2.4 | -2.0 |
Take four scores: 1.2, −1.1, 2.4 and −2.0. Their squares are 1.44, 1.21, 5.76 and 4.00.
SSZ = 1.44 + 1.21 + 5.76 + 4.00 = 12.41. The largest contribution is 5.76 from z = 2.4.
| 1.2² | (−1.1)² | 2.4² | (−2.0)² | SSZ |
|---|---|---|---|---|
| 1.44 | 1.21 | 5.76 | 4.0 | 12.41 |
Squaring maps z and −z to the same contribution. This is appropriate when overall magnitude matters, regardless of whether each result is high or low.
It is not appropriate when direction is the scientific question. Keep the original signed z scores beside SSZ.
For the same four scores, the signed sum is 0.5 and RSZ = 0.5/square-root(4) = 0.25. RSZ looks small because positive and negative departures cancel.
SSZ remains 12.41. The pair therefore says: there is little coherent directional bias, but the combined magnitude deserves attention.
Each squared term’s share of SSZ can be calculated. Here 5.76/12.41 is about 46.4%, while 4/12.41 is about 32.2%.
These shares help prioritise investigation. They do not prove which method step caused the departure.
If the included z scores are independent standard normal variables, their squared sum follows a chi-squared distribution with L degrees of freedom. That model gives reference quantiles for unusually large totals.
The assumptions are substantial. Correlation, non-normality and estimated denominators change the reference behaviour.
Under the simple independent model, degrees of freedom equal the number L of included scores. If four scores are included, the comparison distribution has four degrees of freedom.
Do not use the number of possible analytes when only some valid scores were reported. Record the exact inclusion rule.
Analytes measured by the same preparation, calibration or instrument can share errors. Positive correlation makes simultaneous large scores more likely than the independence model predicts.
A provider may need simulation, covariance modelling or a scheme-specific rule. SSZ can still be descriptive even when a simple chi-squared threshold is unsuitable.
The component scores should be defined consistently. If one uses a target standard deviation and another uses a different uncertainty convention, the squared values may not be comparable.
Audit assigned values, standard deviations for proficiency assessment and transformations before aggregation.
SSZ tends to increase as more scores are included, so totals from participants with different L are not directly comparable without a declared reference.
Selecting only favourable analytes invalidates the interpretation. Prespecify the set or report why values are missing.
Because of squaring, z = 4 contributes 16 while z = 1 contributes 1. This sensitivity is the purpose of SSZ, but it also means one failed analyte can dominate the total.
Always inspect component scores rather than treating SSZ as a complete diagnosis.
Plot two z scores as coordinates. SSZ is the squared distance from the origin: z_1 squared plus z_2 squared. Circles around the origin represent equal SSZ.
This geometry shows why direction changes while magnitude can stay constant. It also links school algebra to multivariate quality assessment.
Use columns for analyte, result, assigned value, proficiency standard deviation, z, z squared and inclusion reason. Sum the squared column only for valid prespecified rows.
Add a contribution percentage column and keep formulas based on unrounded z scores.
Validate finite component scores, square them, sum them and return L, SSZ, signed sum, RSZ and contributions. Preserve analyte identifiers so the diagnostic remains traceable.
Unit-test 1.2, −1.1, 2.4 and −2.0 to obtain SSZ = 12.41 and RSZ = 0.25.
Squaring rounded display values can change the total near a decision threshold. Calculate from full-precision scores and round only for presentation.
State the comparison rule and its inclusivity. A printed value should never conceal how the unrounded decision was made.
SSZ does not identify direction, root cause or which analyte is scientifically most important. It is not automatically comparable across rounds with different score sets.
A small SSZ does not prove absence of systematic problems outside the reported analytes. A large SSZ prompts review rather than supplying a diagnosis.
Multianalyte diagnostics matter in environmental chemistry, food testing, clinical laboratories, geochemistry and nuclear analytical quality assurance. Similar squared-distance ideas appear in engineering residual analysis and machine learning.
The transferable skill is to aggregate without losing the component audit trail.
The RSZ guide detects coherent signed bias. The proficiency-testing z-score guide explains the component score. The z-prime guide shows how assigned-value uncertainty can alter an individual standardised comparison.
SSZ should inherit only scores whose definitions match the declared aggregation.
Students can practise calculating both signed and squared summaries, then explain what each detects. The central lesson is that summaries have blind spots.
Parents can ask to see the component table. A transparent total should invite questions, not end them.
Calculate SSZ for 1, −1, 2 and −2. Reverse every sign and explain why SSZ is unchanged. Add a fifth zero score and discuss what happens to SSZ and L.
Create two four-score sets with the same SSZ but different RSZ values. Explain the patterns.
### Can SSZ be negative? No. Every squared contribution is non-negative.
### Is SSZ the same as RSZ squared? No. The square of a sum includes cross-products; the sum of squares does not.
### Does SSZ always follow chi-squared? Only under the relevant model assumptions, including suitable standard normal component scores and independence.
### Should component scores still be reported? Yes. They preserve direction, analyte identity and diagnostic detail.
Read the official IAEA Analytical Quality in Nuclear Applications report, the current ISO 13528:2022 overview and the NIST chi-square distribution handbook. The worked scores and contribution calculations are original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
SSZ shows why squaring is more than a classroom manipulation. It prevents directional cancellation, creates a geometric measure of total departure and encourages a disciplined return to the individual results that built the total.
Extended practice laboratory
Practice case 1: Square four scores
Setup: Use the article’s SSZ example and change one declared input for case 1. Task: Recalculate from raw values, preserve units and state which mathematical 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 the squares
Setup: Use the article’s SSZ example and change one declared input for case 2. Task: Recalculate from raw values, preserve units and state which mathematical 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 RSZ too
Setup: Use the article’s SSZ example and change one declared input for case 3. Task: Recalculate from raw values, preserve units and state which mathematical 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: Reverse all signs
Setup: Use the article’s SSZ example and change one declared input for case 4. Task: Recalculate from raw values, preserve units and state which mathematical 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 zero score
Setup: Use the article’s SSZ example and change one declared input for case 5. Task: Recalculate from raw values, preserve units and state which mathematical 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: Add a large score
Setup: Use the article’s SSZ example and change one declared input for case 6. Task: Recalculate from raw values, preserve units and state which mathematical 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: Find contributions
Setup: Use the article’s SSZ example and change one declared input for case 7. Task: Recalculate from raw values, preserve units and state which mathematical 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: Change L
Setup: Use the article’s SSZ example and change one declared input for case 8. Task: Recalculate from raw values, preserve units and state which mathematical 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: Check degrees of freedom
Setup: Use the article’s SSZ example and change one declared input for case 9. Task: Recalculate from raw values, preserve units and state which mathematical 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: Inspect cancellation
Setup: Use the article’s SSZ example and change one declared input for case 10. Task: Recalculate from raw values, preserve units and state which mathematical 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: Create equal SSZ patterns
Setup: Use the article’s SSZ example and change one declared input for case 11. Task: Recalculate from raw values, preserve units and state which mathematical 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: Create different RSZ patterns
Setup: Use the article’s SSZ example and change one declared input for case 12. Task: Recalculate from raw values, preserve units and state which mathematical 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: Audit assigned values
Setup: Use the article’s SSZ example and change one declared input for case 13. Task: Recalculate from raw values, preserve units and state which mathematical 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: Audit sigma_pt
Setup: Use the article’s SSZ example and change one declared input for case 14. Task: Recalculate from raw values, preserve units and state which mathematical 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: Check dependence
Setup: Use the article’s SSZ example and change one declared input for case 15. Task: Recalculate from raw values, preserve units and state which mathematical 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: Model positive correlation
Setup: Use the article’s SSZ example and change one declared input for case 16. Task: Recalculate from raw values, preserve units and state which mathematical 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: Model negative correlation
Setup: Use the article’s SSZ example and change one declared input for case 17. Task: Recalculate from raw values, preserve units and state which mathematical 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: Handle missing analytes
Setup: Use the article’s SSZ example and change one declared input for case 18. Task: Recalculate from raw values, preserve units and state which mathematical 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: Prespecify inclusion
Setup: Use the article’s SSZ example and change one declared input for case 19. Task: Recalculate from raw values, preserve units and state which mathematical 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: Use unrounded scores
Setup: Use the article’s SSZ example and change one declared input for case 20. Task: Recalculate from raw values, preserve units and state which mathematical 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: Plot component squares
Setup: Use the article’s SSZ example and change one declared input for case 21. Task: Recalculate from raw values, preserve units and state which mathematical 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: Investigate the largest term
Setup: Use the article’s SSZ example and change one declared input for case 22. Task: Recalculate from raw values, preserve units and state which mathematical 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 root-cause overclaim
Setup: Use the article’s SSZ example and change one declared input for case 23. Task: Recalculate from raw values, preserve units and state which mathematical 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 multianalyte conclusion
Setup: Use the article’s SSZ example and change one declared input for case 24. Task: Recalculate from raw values, preserve units and state which mathematical 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.