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eduKateSG · Why Mathematics? · Sn Scale Estimation, Nested Medians and Asymmetric Robust Spread
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 4–7
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 8–13
Check assumptions
See where design and calibration matter.Route 4 · Chapters 14–18
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 19–24
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a distribution is asymmetric or contains a few distant measurements? A familiar standard deviation can be pulled by extreme values, while a centre-based measure such as MAD describes distances from one median. The Sn scale estimator takes another route: it finds a typical distance around every observation and then takes a typical value of those typical distances.
Sn brings together absolute differences, nested medians, robustness and model calibration. It offers a clear lesson in how a statistic’s architecture determines what changes can influence it.
Did you know? Sn uses a median inside another median. That nested design lets local distance patterns vote before the dataset produces one final scale.
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
Scale describes how widely quantitative values vary in their original unit. If measurements are in micrometres, Sn is in micrometres.
The method is designed to remain informative when a minority of observations are unusually distant. It is an estimate of spread, not an automatic outlier detector.
NIST defines Sn = 1.1926 × median_i{median_j|x_i − x_j|}. For each observation i, calculate its absolute distance from every observation j, including itself, and take the median of those distances. Then take the median across the n inner medians.
The factor 1.1926 makes the statistic consistent for scale under a normal model in the stated convention. Software may apply finite-sample corrections, so reports should name the implementation.
| Data | Inner medians | Outer median |
|---|---|---|
| 10, 10, 11, 12, 12, 13, 14, 40 | 2, 2, 1, 1.5, 1.5, 1.5, 2.5, 28 | 1.75 |
Consider 10, 10, 11, 12, 12, 13, 14 and 40. For the first 10, the sorted distances are 0, 0, 1, 2, 2, 3, 4 and 30. Their median is the average of the fourth and fifth values: 2.
Repeating this step gives inner medians 2, 2, 1, 1.5, 1.5, 1.5, 2.5 and 28.
| Outer median | Consistency factor | Sn |
|---|---|---|
| 1.75 | 1.1926 | 2.08705 |
Sort the inner medians: 1, 1.5, 1.5, 1.5, 2, 2, 2.5 and 28. The outer median is (1.5 + 2)/2 = 1.75.
Therefore Sn = 1.1926 × 1.75 = 2.08705 units before any optional finite-sample correction.
The observation 40 has an inner median of 28, but that is only one of eight inner summaries. Its very large value sits at the end of the outer list and does not determine the middle two positions.
Changing 40 to 400 makes its local median much larger while the other central distance patterns remain similar. Sn is resistant because the final median limits one observation’s influence.
An ordinary average lets every magnitude contribute directly. A median depends mainly on order around its middle position. Sn applies that resistance twice.
The inner median describes the neighbourhood around each point. The outer median asks what a typical neighbourhood looks like.
NIST reports a 50% breakdown point and a bounded influence function for Sn. It also reports about 58% efficiency under a normal model, higher than the approximately 37% quoted for MAD.
These are model properties, not guarantees that any sample is representative or free of measurement problems.
Both Sn and Qn use pairwise distances and avoid defining spread around one centre. Qn selects a particular ordered distance from all unordered pairs. Sn constructs one median distance per observation and then takes their median.
The Qn guide explains the order-statistic route. Sn’s nested medians reveal a different robust mechanism.
MAD is median|x_i − median(x)|. It is compact and highly resistant, but it measures distance from a central location.
Sn measures typical distances between observations. NIST notes that this makes Sn suitable for asymmetric as well as symmetric distributions, though interpretation still requires a model and context.
Standard deviation is essential when squared deviations, variance decomposition or a normal likelihood match the task. Sn is useful when tail contamination would give a misleading first picture of spread.
Calculating both can expose sensitivity. A large gap invites inspection; it does not itself prove contamination.
Adding the same constant to every observation leaves every pairwise distance unchanged, so Sn is translation invariant. Multiplying every value by a positive factor multiplies Sn by that factor.
These invariants are powerful checks in code and spreadsheets.
Repeated measurements create zero distances. Coarse rounding can reduce inner medians and may make Sn zero in a highly tied sample.
A zero scale can describe limited resolution rather than perfect physical constancy. Record resolution and rounding rules.
With an even number of distances, the ordinary median averages the two middle values. Different finite-sample corrections can then affect the final result.
Publish n, the correction convention and the software version, especially when the result informs a decision.
Students can build one row of distances for each observation and highlight the middle pair. Colour the eight inner medians, sort them and find the outer median.
Then replace 40 by 400 and identify what changes. The nested structure becomes visible rather than mysterious.
Construct an n-by-n absolute-distance matrix, including the zero diagonal. Take the row median for each observation and then the median of that column.
Label rows and columns with observation IDs. A traceable matrix is easier to audit than a single formula.
For each value, generate all n absolute distances, calculate their median, and take the median of the n results. Multiply by the stated consistency factor and optional correction.
Unit-test the example to obtain inner median 1.75 and uncorrected Sn 2.08705.
Sn does not identify which observation is wrong, estimate a confidence interval automatically or make biased sampling representative.
Robust does not mean assumption-free. The population, dependence, censoring and resolution still matter.
Robust scale estimates support exploratory work in metrology, sensor analysis, environmental monitoring, finance and engineering. They are useful when a few exceptional values may be genuine or may require investigation.
The transferable skill is to match the scale estimator to the decision and preserve the raw evidence.
The modified z-score guide uses MAD to flag potential outliers. The Algorithm A guide explains an iterative robust consensus method. Sn instead estimates scale directly through nested pairwise medians.
Students should explain the two median stages in words before multiplying by 1.1926. This builds understanding of why the statistic resists a distant point.
Parents can encourage questions about units, resolution and what happened to unusual observations after they were flagged.
Recalculate Sn after replacing 14 with 15. Add 100 to every value and predict the result. Double every observation and verify the scale change.
Create a tied dataset and explain how zero distances influence the inner medians.
### Is Sn an average of all pairwise distances? No. It uses a median for each observation and then a median of those medians.
### Does Sn find outliers? No. It estimates robust scale; separate evidence is needed to investigate observations.
### Why 1.1926? It is a consistency factor for scale under a normal model in the cited convention.
### Is Sn always better than standard deviation? No. Choose according to the model, contamination risk and reporting purpose.
Read the official NIST Sn scale documentation, the original research Alternatives to the Median Absolute Deviation and the NIST Qn documentation. The worked calculations are original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
Sn turns local distances into a robust global scale. Its nested medians show students how careful mathematical structure can reduce tail influence while keeping every observation visible for investigation.
Extended practice laboratory
Practice case 1: Build one distance row
Setup: Use the article’s Sn 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 every row median
Setup: Use the article’s Sn 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 the outer median
Setup: Use the article’s Sn 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: Apply 1.1926
Setup: Use the article’s Sn 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: Move the far value
Setup: Use the article’s Sn 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: Move a central value
Setup: Use the article’s Sn 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 data
Setup: Use the article’s Sn 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: Double all data
Setup: Use the article’s Sn 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: Convert units
Setup: Use the article’s Sn 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: Add a tie
Setup: Use the article’s Sn 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 scale
Setup: Use the article’s Sn 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 Qn
Setup: Use the article’s Sn 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 MAD
Setup: Use the article’s Sn 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 standard deviation
Setup: Use the article’s Sn 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: Inspect asymmetry
Setup: Use the article’s Sn 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 finite correction
Setup: Use the article’s Sn 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: Audit software version
Setup: Use the article’s Sn 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 values
Setup: Use the article’s Sn 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 row medians
Setup: Use the article’s Sn 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: Check resolution
Setup: Use the article’s Sn 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: Use small n
Setup: Use the article’s Sn 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 contamination
Setup: Use the article’s Sn 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: Explain breakdown
Setup: Use the article’s Sn 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 robust-scale conclusion
Setup: Use the article’s Sn 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.