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eduKateSG · Why Mathematics? · Percentage Difference Scores, Assigned Values and Relative Error
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 19–26
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 measurement must be compared with an assigned value? The raw difference preserves the unit, while percentage difference expresses that deviation relative to the reference. The denominator gives the percentage its meaning.
In proficiency testing, the percentage difference score is simple enough to calculate by hand and subtle enough to misuse. It requires a defensible assigned value, matching units, a non-zero denominator and a separate decision rule.
Did you know? A result can be 4% above an assigned value yet use 80% of a practical error allowance. The two percentages have different denominators and answer different questions.
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 19–26 · Study and verify
Let x_i be a participant result and x_ref the assigned value. The signed difference D_i = x_i − x_ref shows the deviation in the original unit.
The percentage difference asks how large that deviation is compared with x_ref.
NIST’s ISO 13528 implementation defines D_i% = 100(x_i − x_ref)/x_ref. The multiplication by 100 converts the ratio to percentage points.
The sign indicates direction: positive is above the reference and negative is below it, assuming a positive reference.
| Participant result | Assigned value | Signed difference |
|---|---|---|
| 104 mg/kg | 100 mg/kg | 4 mg/kg |
Suppose x_i = 104 mg/kg and x_ref = 100 mg/kg. The signed difference is 4 mg/kg.
D_i% = 100 × 4/100 = 4%. The result is four percent above the assigned value.
| Formula | Result |
|---|---|
| 100×(104−100)/100 | 4% |
If x_i = 96 mg/kg with the same reference, D_i = −4 mg/kg and D_i% = −4%.
The equal magnitudes occur because both results are compared with the same fixed denominator of 100 mg/kg.
A 4-unit difference is 4% of 100 but 40% of 10. The raw difference alone cannot express relative size.
Before interpreting any percentage, ask “percentage of what?” Here the answer is the assigned value.
CHAPTER 6 OF 26 · Reproduce the example
6. Percentage difference versus percent error
The phrase “percent error” is used inconsistently in classrooms and technical work. Some sources remove the sign; others preserve it.
Use the explicit formula and the term defined by the scheme. Report both signed D_i and signed D_i% when direction matters.
The proportion of allowed limit score divides by a practical allowance delta_E, not by x_ref. With x_i = 104, x_ref = 100 and delta_E = 5, D_i% is 4% but PA is 100×4/5 = 80%.
D_i% describes relative deviation from the reference. PA describes how much of an allowed error has been used.
A proficiency z score divides by the standard deviation for proficiency assessment. It expresses deviation in target-standard-deviation units.
D_i% uses the assigned value as denominator. Neither quantity should be substituted for the other without a documented relationship.
NIST explains that D_i or D_i% can be compared with a pre-defined acceptable error delta_E. If the rule is stated in original units, the equivalent percentage boundary depends on x_ref.
The formula does not create the tolerance. Fitness for purpose, regulation or method requirements must justify it.
If x_ref = 0, D_i% is undefined. A very small reference can also produce an enormous percentage from a modest absolute difference.
Use an absolute tolerance, detection-limit framework or another documented metric when zero is scientifically meaningful.
For quantities that can be negative, denominator sign complicates intuitive “above” and “below” language. The numerical formula can reverse sign relationships.
Choose a metric designed for the quantity and explain the baseline. Many concentration applications avoid this issue because assigned values are positive.
x_i and x_ref must use the same unit. Converting both from mg/kg to micrograms/kg multiplies numerator and denominator by the same factor, leaving D_i% unchanged.
Converting only one value creates a false result. Keep units beside every input.
An assigned value may come from formulation, a certified reference, a reference method, expert laboratories or robust participant consensus. Each route carries uncertainty and possible dependence.
A precise percentage does not repair a weak reference. Audit how x_ref was established.
The basic percentage difference formula does not combine the participant’s uncertainty with assigned-value uncertainty. Those uncertainties may still matter to interpretation.
Use z-prime, zeta, En or another defined method when the decision question requires explicit uncertainty.
A true value of 4.96% may display as 5.0%. If 5% is a decision boundary, classification should use unrounded inputs according to the declared rule.
Store enough digits to reproduce the decision and state whether equality is included.
Give students three references—10, 100 and 1,000—with the same +4 difference. Calculate the three percentages and discuss why they change.
Then set the reference to zero. The failed division becomes a concrete lesson about domain restrictions.
Use columns for result, assigned value, units, signed difference and percentage difference. Add validation that units match and x_ref is non-zero.
Keep the assigned-value source and any tolerance in adjacent metadata fields.
Check finite numeric inputs and matching units, reject zero reference values and compute from unrounded numbers. Return both D_i and D_i%.
Unit-test 104 and 100 to obtain 4 units and 4%. Test common scaling to verify invariance.
Percentage difference is not a probability of correctness, not an uncertainty interval and not automatically a pass/fail score.
Comparisons across analytes can also mislead if their acceptable relative errors differ. Context decides whether 4% is small enough.
Relative differences appear in analytical laboratories, calibration, manufacturing, environmental measurement, finance and data quality. The same denominator discipline helps students compare change in everyday life.
Professionals add domain tolerances, uncertainty and traceability; the arithmetic is the beginning of the decision.
The proportion-of-allowed-limit guide uses a practical tolerance denominator. The z-score guide uses a proficiency standard deviation. The log-transformed z-score guide treats positive multiplicative error through logarithms.
Reading them together shows how denominator and model determine meaning.
Students should name the numerator and denominator in words before calculating. This prevents many percentage errors.
Parents can ask whether the reference is trustworthy, whether zero is possible and what rule turns the percentage into an action.
Calculate D and D% for 49 compared with 50. Convert both to a different unit and verify invariance. Compare a +2 difference around references of 10 and 200.
Explain why percentage difference is undefined at a zero reference and propose an absolute comparison.
### Is percentage difference always positive? Not in this signed definition. The sign preserves direction.
### Does 4% automatically pass? No. The provider must define an acceptable limit for the intended use.
### Can it be used when the reference is zero? No. The denominator makes the score undefined.
### Is it the same as PA? No. PA divides by an allowed error, while D_i% divides by the assigned value.
Read the official NIST percentage-difference documentation, the current ISO 13528:2022 overview and the Eurachem proficiency-testing guide. The worked values are original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
Percentage difference teaches a lasting mathematical lesson: a ratio is an argument about scale. Once the denominator, reference quality and decision purpose are explicit, a simple percentage becomes a clear and auditable comparison.
Extended practice laboratory
Practice case 1: Compute a positive score
Setup: Use the article’s percentage-difference 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: Compute a negative score
Setup: Use the article’s percentage-difference 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: Change the reference
Setup: Use the article’s percentage-difference 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: Keep the same raw difference
Setup: Use the article’s percentage-difference 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: Convert all units
Setup: Use the article’s percentage-difference 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: Convert one input wrongly
Setup: Use the article’s percentage-difference 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: Set reference to zero
Setup: Use the article’s percentage-difference 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: Use a tiny reference
Setup: Use the article’s percentage-difference 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: Inspect a negative reference
Setup: Use the article’s percentage-difference 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: Compare absolute difference
Setup: Use the article’s percentage-difference 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: Compare PA
Setup: Use the article’s percentage-difference 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: Compare z
Setup: Use the article’s percentage-difference 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 value
Setup: Use the article’s percentage-difference 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 tolerance
Setup: Use the article’s percentage-difference 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 uncertainty separately
Setup: Use the article’s percentage-difference 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: Round near a boundary
Setup: Use the article’s percentage-difference 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: Test equality at a boundary
Setup: Use the article’s percentage-difference 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: Preserve sign
Setup: Use the article’s percentage-difference 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: Preserve units
Setup: Use the article’s percentage-difference 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: Handle missing values
Setup: Use the article’s percentage-difference 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 repeated percentages
Setup: Use the article’s percentage-difference 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 persistent direction
Setup: Use the article’s percentage-difference 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 probability language
Setup: Use the article’s percentage-difference 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 relative-error conclusion
Setup: Use the article’s percentage-difference 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.