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eduKateSG · Why Mathematics? · Modified z Scores, Median Absolute Deviation and Potential Outliers
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 one observation looks suspicious? A graph can attract attention, but a reproducible investigation needs a stated rule. The modified z score measures distance from the median in units based on the median absolute deviation, reducing the masking that can occur when an extreme observation pulls both the mean and standard deviation.
This method connects median, absolute value, scaling and cautious evidence language. It labels potential outliers for investigation; it does not declare mistakes automatically.
Did you know? NIST recommends the phrase “potential outlier” for an absolute modified z score greater than 3.5. That wording preserves the next step: check the observation and its context.
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
The ordinary z score uses distance from the mean divided by standard deviation. When the same extreme value helps determine both quantities, it can partly hide itself.
The modified z score uses a more resistant centre and spread. It asks how unusual an observation is relative to the median structure of the sample.
For observation x_i, sample median x-tilde and MAD = median|x_i − x-tilde|, NIST gives M_i = 0.6745(x_i − x-tilde)/MAD.
The factor 0.6745 calibrates MAD to the normal distribution. An equivalent form divides the signed distance by MAD/0.6745.
| Data | Median | Absolute deviations | MAD |
|---|---|---|---|
| 9, 10, 10, 11, 30 | 10 | 1, 0, 0, 1, 20 | 1 |
Consider 9, 10, 10, 11 and 30. The median is 10. Absolute deviations from the median are 1, 0, 0, 1 and 20; after sorting, their median is 1, so MAD = 1.
For 30, M = 0.6745(30 − 10)/1 = 13.49. Its absolute score is well above 3.5, so the NIST rule labels it a potential outlier.
| Calculation | Modified z | Interpretation |
|---|---|---|
| 0.6745×(30−10)/1 | 13.49 | Potential outlier under |M| > 3.5 rule |
For 9, M = −0.6745. Each 10 has M = 0, and 11 has M = 0.6745. These calculations keep direction through the sign.
The table of all scores is more informative than reporting only the largest one. It shows the reference structure against which 30 was judged.
Changing 30 to 300 leaves the median at 10 and MAD at 1. The unusual value’s modified score grows instead of dragging the reference centre and spread far towards itself.
This is resistance. It does not guarantee perfect detection when several unusual values cluster together or when the distribution is naturally skewed.
For a standard normal distribution, the median absolute deviation is about 0.6745. Multiplying the numerator by 0.6745 makes the score comparable to standard-deviation units under that model.
The factor is a calibration constant, not a probability. A score of 3.5 does not mean a 3.5% chance of error.
NIST cites the recommendation that |M_i| greater than 3.5 be labelled a potential outlier. A scheme should state whether the boundary itself is included and calculate from unrounded values.
The rule is an exploratory flag. Domain standards may require another test or may forbid automatic exclusion.
If at least half the observations equal the median, MAD can be zero. Division then fails even though some values differ.
Do not add an arbitrary tiny denominator. Report that the rule is undefined for the data resolution and use a documented alternative or inspect the exact frequency structure.
Rounded readings can create many ties and a zero or very small MAD. A large modified z score may then reflect coarse recording precision as much as a physical anomaly.
Record instrument resolution, data entry rules and any censoring such as “below detection limit.”
A long right tail may be genuine rather than contaminated. A symmetric two-sided cutoff can flag valid high values more often than low values.
Transformations, one-sided models or distribution-specific methods may be more suitable. Always inspect a histogram, dot plot or time sequence.
Several unusual observations can change the median and MAD or form a cluster that appears ordinary relative to itself. No single score solves every masking and swamping problem.
Use source records, replicate measurements and process knowledge. Robust scores strengthen an investigation when combined with independent evidence.
Calculate the median and MAD from a clearly defined analysis set. Removing inconvenient observations before the calculation changes the reference structure and can make the method circular.
Document missingness, exclusions and duplicate handling before interpreting scores.
Plot 9, 10, 10, 11 and 30 on a number line. Calculate the ordinary mean and median, then replace 30 with 300 and compare how each centre changes.
Next calculate MAD and modified z scores. Students can explain in words why the reference structure stays stable.
Use one cell for the median, a column for absolute deviations, one cell for MAD and a final column for signed modified scores. Protect references carefully so copied formulas use the same median and MAD.
Add a flag column based on the declared threshold, but keep the raw observation and score visible. Never replace a flagged value automatically.
Compute the median after applying the documented missing-data rule, compute absolute deviations and their median, then handle MAD = 0 explicitly. Return all scores and a Boolean potential-outlier flag.
Unit-test the five-value example: median 10, MAD 1 and M_30 = 13.49. Test translation invariance and invariance to multiplying every value by a positive constant.
Adding the same constant to all observations leaves each modified z score unchanged. Multiplying every value by the same positive factor also leaves scores unchanged because numerator and MAD scale together.
The score is dimensionless, but the raw difference and MAD should retain their unit in the audit record.
A potential outlier is not necessarily a recording error. It could be a rare valid event, a process shift, a different population or a scientific discovery.
Conversely, a value below 3.5 is not certified correct. The rule has limited power in small or complicated datasets.
CHAPTER 18 OF 24 · Explore applications
18. Applications in science, technology and education
Modified z scores support preliminary checks in laboratory measurements, sensor streams, manufacturing records and data cleaning. In education data, they can prompt review of scores or timings while protecting students from automatic conclusions based on one number.
The transferable skill is to create an auditable flag and then seek context.
The Mathematics Learning Hub connects this topic with pairwise-distance scale and other robust approaches. The proficiency-testing z-score guide uses an assigned value and a proficiency standard deviation, so its denominator and purpose differ from this within-sample diagnostic.
The Algorithm A guide explains why robust reference estimates require a declared procedure.
Students should write “potential outlier under the stated rule,” then list the evidence needed next. This is stronger reasoning than deleting a value because it looks large.
Parents can ask how the centre and spread were chosen, whether the data are rounded and whether a real-world explanation was checked.
Calculate modified z scores for 4, 5, 5, 6 and 20. Replace 20 with 200 and compare the reference quantities. Construct a dataset with MAD = 0 and explain the computational problem.
Create a right-skewed dataset and discuss why a symmetric cutoff may require caution.
### Is a modified z score a proficiency-testing z score? No. It uses a sample median and MAD rather than a scheme’s assigned value and proficiency standard deviation.
### Should every value above 3.5 be deleted? No. It should be investigated and handled under a documented scientific rule.
### What if MAD is zero? The formula is undefined. Report the issue and use a justified alternative.
### Does the sign matter? Yes. It shows whether the observation is below or above the median.
Use the official NIST outlier-detection handbook, the NIST MAD definition and NIST Internal Report 8526. The worked example is original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
Modified z scores turn an impression into a reproducible question. The mathematics provides a resistant flag; responsible analysis supplies the investigation, context and decision.
Extended practice laboratory
Practice case 1: Find the median
Setup: Use the article’s modified-z 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 absolute deviations
Setup: Use the article’s modified-z 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 MAD
Setup: Use the article’s modified-z 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: Score the high value
Setup: Use the article’s modified-z 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: Score the low value
Setup: Use the article’s modified-z 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: Replace 30 by 300
Setup: Use the article’s modified-z 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: Translate all values
Setup: Use the article’s modified-z 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: Scale all values
Setup: Use the article’s modified-z 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: Apply the 3.5 rule
Setup: Use the article’s modified-z 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: Check equality at 3.5
Setup: Use the article’s modified-z 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 zero MAD
Setup: Use the article’s modified-z 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: Handle ties
Setup: Use the article’s modified-z 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: Inspect skewness
Setup: Use the article’s modified-z 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: Compare ordinary z
Setup: Use the article’s modified-z 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: Plot the observations
Setup: Use the article’s modified-z 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: Audit rounding
Setup: Use the article’s modified-z 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: Handle missing data
Setup: Use the article’s modified-z 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: Check units
Setup: Use the article’s modified-z 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: Investigate a flag
Setup: Use the article’s modified-z 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: Preserve the raw value
Setup: Use the article’s modified-z 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: Test multiple anomalies
Setup: Use the article’s modified-z 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: Avoid automatic deletion
Setup: Use the article’s modified-z 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: Record the rule
Setup: Use the article’s modified-z 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 potential-outlier conclusion
Setup: Use the article’s modified-z 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.