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eduKateSG · Why Mathematics? · Quartile Coefficient of Dispersion, Relative Spread and Positive Scales
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 two positive-valued datasets need a relative, quartile-based comparison of spread? The quartile coefficient of dispersion divides the interquartile range by the sum of the lower and upper quartiles. The result is dimensionless and focuses on the middle half.
The method connects quartiles, ratios and scale invariance. It can be a robust alternative to the coefficient of variation when a positive scale and meaningful quartiles support the comparison.
Did you know? Multiplying every observation by 1,000 changes Q1 and Q3 but leaves their difference-to-sum ratio unchanged.
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
An interquartile range is expressed in the original unit. Comparing IQRs across variables with very different levels can be difficult.
The quartile coefficient creates a relative measure using the same two quartiles in numerator and denominator.
NIST defines QCD = (Q3−Q1)/(Q1+Q3), where Q1 is the 25th percentile and Q3 the 75th percentile.
The statistic has been suggested as a robust alternative to the coefficient of variation.
| Q1 | Q3 | IQR | Quartile sum |
|---|---|---|---|
| 10 | 14 | 4 | 24 |
Suppose Q1 = 10 units and Q3 = 14 units. The interquartile range is 4 units and the quartile sum is 24 units.
QCD = 4/24 = 1/6 ≈ 0.1667, or 16.67% if expressed as a percentage.
| Formula | Ratio | Percentage |
|---|---|---|
| (14−10)/(14+10) | 1/6 | 16.67% |
The numerator measures middle-half width. The denominator represents the combined level of the two quartile boundaries.
The ratio is dimensionless, but it is not a probability.
Multiply both quartiles by a positive factor a. The factor appears in numerator and denominator and cancels.
This makes QCD useful for positive measurements recorded in different units.
Adding a constant changes Q1+Q3 but leaves Q3−Q1 unchanged. QCD therefore changes under translation.
That is appropriate for ratio-scale quantities with a meaningful zero, but problematic when zero is arbitrary.
If Q1+Q3 is zero, the statistic is undefined. If the sum is near zero, the ratio can become unstable.
Negative or sign-changing quantities require special care because the denominator may not represent a meaningful level.
IQR = Q3−Q1 stays in the original unit and is translation invariant. QCD normalises by Q1+Q3 and becomes unitless.
Choose absolute or relative spread according to the comparison question.
The coefficient of variation uses standard deviation divided by mean. QCD uses quartiles instead, reducing dependence on tails and the mean.
Both need a meaningful positive scale and can misbehave near zero.
Software uses different percentile algorithms. With small datasets, Q1 and Q3 can differ across methods, changing QCD.
Report the quantile convention or software version.
Repeated values can make Q1 equal Q3, producing QCD zero even when observations outside the middle half vary.
Show the full distribution and retained resolution.
QCD values can compare relative central spread across groups measured on compatible positive scales. Sample size and uncertainty still matter.
A difference in sample estimates is not automatically a meaningful population difference.
Draw two box plots with quartiles 10 and 14, then 100 and 140. Calculate QCD for both and verify equality.
Next add 10 to every value and explain why the ratio changes.
Calculate Q1 and Q3 under a declared quantile type, validate a non-zero meaningful denominator, then compute the difference-to-sum ratio.
Keep quartiles visible beside the final coefficient.
Return Q1, Q3, IQR, quartile sum and QCD. Reject or flag near-zero denominators and non-finite input.
Unit-test Q1=10 and Q3=14 to obtain 0.1666667.
Calculate from unrounded quartiles. Rounding both boundaries first can distort a small IQR.
Store sufficient precision and round the display last.
QCD does not describe tails outside the quartiles, prove robustness against every contamination pattern or create comparability across unrelated quantities.
It is not translation invariant and should not be used blindly for temperatures on arbitrary-zero scales.
Relative quartile spread can support comparisons in quality analysis, environmental measurements, finance and skewed positive data. Similar dimensionless ratios help engineers compare systems at different scales.
The transferable skill is checking whether zero and scaling are scientifically meaningful.
The midmean guide averages values in the interquartile region. The Mathematics Learning Hub connects this relative measure with pairwise-gap and absolute-scale methods. The Qn guide estimates robust absolute scale.
Students should say what cancels and why the result has no unit. They should also test what happens after adding a constant.
Parents can ask whether the data have a meaningful zero and whether quartile conventions were documented.
Calculate QCD for Q1=20 and Q3=30. Multiply both by 100. Add 5 to both and compare. Explore what happens when Q1=-Q3.
Compare QCD and IQR for two positive datasets with different levels.
### Is QCD a percentage? It is a ratio and may be multiplied by 100 for percentage presentation.
### Does changing units affect it? Positive multiplicative unit changes cancel.
### Does adding a constant leave it unchanged? No. The denominator changes while the IQR does not.
### Can it be used near zero? It can become unstable or undefined, so another metric may be needed.
Read the official NIST quartile coefficient documentation, the NIST measures-of-scale handbook and the referenced Bonett confidence-interval article record. The worked ratio is original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
The quartile coefficient of dispersion shows how a ratio can turn central spread into a unitless comparison. Its usefulness depends on a meaningful positive scale and transparent quartile rules.
Extended practice laboratory
Practice case 1: Find Q1
Setup: Use the article’s QCD 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: Find Q3
Setup: Use the article’s QCD 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: Find IQR
Setup: Use the article’s QCD 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: Find quartile sum
Setup: Use the article’s QCD 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: Calculate QCD
Setup: Use the article’s QCD 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: Express as percent
Setup: Use the article’s QCD 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: Scale all values
Setup: Use the article’s QCD 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: Add a constant
Setup: Use the article’s QCD 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: Change units
Setup: Use the article’s QCD 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: Use near-zero quartiles
Setup: Use the article’s QCD 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: Use negative quartiles
Setup: Use the article’s QCD 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 IQR
Setup: Use the article’s QCD 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: Compare CV
Setup: Use the article’s QCD 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: Change quantile type
Setup: Use the article’s QCD 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: Add ties
Setup: Use the article’s QCD 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: Use small n
Setup: Use the article’s QCD 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: Preserve raw values
Setup: Use the article’s QCD 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 data
Setup: Use the article’s QCD 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: Plot a box plot
Setup: Use the article’s QCD 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: Compare groups
Setup: Use the article’s QCD 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: Estimate uncertainty
Setup: Use the article’s QCD 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: Check meaningful zero
Setup: Use the article’s QCD 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 QCD 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-spread conclusion
Setup: Use the article’s QCD 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.