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eduKateSG · Why Mathematics? · Log-Transformed z Scores, Multiplicative Error and Proportional Precision
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
Some proficiency-test results vary proportionally rather than by a constant number of units. A result twice as large may naturally show roughly twice the absolute spread. Log-transformed z scores convert ratios into differences, making multiplicative variation easier to assess.
This topic shows why mathematics is important in science and everyday reasoning: logarithms turn “times as large” into an additive distance while preserving direction.
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
An ordinary z score uses x – X. For positive quantities with proportional error, the ratio x/X may be more stable across concentration levels.
Taking logarithms gives log(x) – log(X) = log(x/X). The score divides that log difference by a declared standard deviation on the log scale.
Let x = 120, assigned value X = 100 and sigma_log = 0.10 natural-log units. The ratio is 1.2.
The numerator is ln(1.2) ≈ 0.18232. Therefore z_log = 0.18232/0.10 ≈ 1.8232.
| x | X | Ratio x/X | sigma_log |
|---|---|---|---|
| 120 | 100 | 1.2 | 0.1 |
If x > X, the ratio exceeds one and its log is positive. If 0 < x < X, the ratio lies between zero and one and its log is negative.
Equal reciprocal ratios have opposite logs: ln(1.2) = -ln(1/1.2). This produces symmetric treatment of proportional high and low deviations.
| Step | Calculation | Result |
|---|---|---|
| Log ratio | ln(1.2) | 0.18232 |
| z_log | 0.18232/0.10 | 1.8232 |
CHAPTER 4 OF 23 · Reproduce the example
4. Why ordinary percentages are not identical
A result 20% high has ratio 1.2. A result 20% low has ratio 0.8. Their log magnitudes are ln(1.2)=0.1823 and |ln(0.8)|=0.2231, so they are not multiplicative reciprocals.
The exact reciprocal of 1.2 is about 0.8333, or 16.67% low.
The assessment standard deviation must be defined on the same log scale and aligned with the proficiency purpose. It can arise from a proportional performance requirement or a transformed dispersion model.
Do not take logs of results while leaving an untransformed sigma in the denominator.
For a lognormal model, a coefficient of variation CV can be related to log-scale standard deviation by sigma_log = square root of ln(1 + CV squared). With CV = 0.10, sigma_log ≈ 0.09975.
This relationship is model-based. State it rather than treating 10% and 0.10 log units as exactly identical.
Ordinary logarithms are undefined at zero and for negative measured values. Censored, zero or background-corrected results need a method designed for them.
Adding an arbitrary constant merely to enable logs changes the scientific meaning.
Natural logs and base-10 logs differ by a constant factor. If both numerator and sigma are expressed in the same base, the z score is unchanged.
Mixing bases produces a false scale.
The assigned value must be positive and established by the scheme's method. Its uncertainty can still matter, and a log-scale analogue of z-prime may be appropriate under a documented procedure.
Transformation does not eliminate reference uncertainty.
A log-score interval can be translated back to multiplicative factors. If sigma_log = 0.10, one assessment SD corresponds to multiplying or dividing by exp(0.10) ≈ 1.1052.
Back-transformed explanations help readers connect logs to original units.
Logs can stabilise relative spread, reduce right skew and turn multiplicative effects into additive ones. They are common for concentrations spanning orders of magnitude.
They are inappropriate when additive error is scientifically natural or when values cross zero.
Plot standard deviation against mean on the original scale. Proportional growth suggests a log model. Then inspect the transformed distribution and residuals.
Choose transformation from evidence and mechanism, not from a desire for comfortable scores.
Compare ratios 0.5, 0.8, 1, 1.25 and 2. Compute their natural logs and plot ratio against log ratio. Notice how reciprocal pairs become symmetric around zero.
Then calculate scores with one declared sigma_log.
Validate x>0 and X>0. Calculate ratio, log ratio, sigma_log and score in separate columns. Preserve the original values and units.
Add a clear note stating log base and how sigma_log was obtained.
Reject non-positive inputs, enforce one log base and return both log score and multiplicative ratio. Test x=X to obtain zero and reciprocal ratios to obtain opposite scores.
Keep full precision before classification.
For small relative deviations, ln(x/X) is approximately (x-X)/X. The log score can therefore resemble a relative-error score near the assigned value.
At larger deviations the nonlinear difference becomes important.
Students can connect indices, logarithms, ratios and statistical standardisation. Ask why doubling and halving sit equally far from one on the log scale.
Parents can encourage back-transformation so the answer remains interpretable.
Log-scale performance analysis appears in chemistry, environmental science, microbiology and any field with positive quantities spanning wide ranges. The broader benefit is learning when additive and multiplicative models differ.
Explore the Mathematics Learning Hub and How Mathematics Works.
The HorRat guide uses a power law for concentration-dependent precision. The PT z-score guide explains ordinary standardisation.
Calculate z_log for x=80, X=100 and sigma_log=0.10. Find the reciprocal low result corresponding to x=120. Convert the calculation to base-10 logs with a matching sigma. Try x=0 and explain why the method fails.
### Is a log score just a percentage score? No. It standardises a logarithm of a ratio; percentages are only approximately related for small deviations.
### Can negative results be logged? Not with the ordinary real logarithm. Use a scientifically appropriate alternative.
### Which log base should I use? Any base works if the numerator and scale use the same base and the scheme documents it.
### Does transformation remove outliers? No. It changes geometry and may reduce right skew, but unusual results still need investigation.
Use the Eurachem proficiency-testing resource list, which identifies the Analytical Methods Committee technical brief on log-transformed z scores, the Eurachem PT guide and official ISO 13528:2022. The worked numbers are original.
Log-transformed z scores make proportional error visible. Ratios become additive distances, reciprocal changes become symmetric, and the scoring scale can follow the real measurement mechanism.
Extended practice laboratory
Practice case 1: Score a 20 percent high result
Setup: Use the article’s log-z 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: Score a reciprocal low result
Setup: Use the article’s log-z 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: Use base ten logs
Setup: Use the article’s log-z 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: Match log scale
Setup: Use the article’s log-z 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: Convert CV to sigma_log
Setup: Use the article’s log-z 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: Approximate small deviations
Setup: Use the article’s log-z 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: Try zero
Setup: Use the article’s log-z 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: Try a negative value
Setup: Use the article’s log-z 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: Shift units improperly
Setup: Use the article’s log-z 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: Scale both x and X
Setup: Use the article’s log-z 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: Double the ratio
Setup: Use the article’s log-z 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: Use additive error instead
Setup: Use the article’s log-z 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: Plot SD versus mean
Setup: Use the article’s log-z 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: Inspect transformed residuals
Setup: Use the article’s log-z 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: Back-transform one SD
Setup: Use the article’s log-z 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: Audit assigned value
Setup: Use the article’s log-z 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: Audit sigma_log
Setup: Use the article’s log-z 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: Compare ordinary z
Setup: Use the article’s log-z 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 HorRat
Setup: Use the article’s log-z 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: Check rounding
Setup: Use the article’s log-z 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: Preserve original units
Setup: Use the article’s log-z 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: Test a large deviation
Setup: Use the article’s log-z 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: Document log base
Setup: Use the article’s log-z 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 multiplicative-model conclusion
Setup: Use the article’s log-z 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.