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eduKateSG · Why Mathematics? · Shortest-Half Midmean, Compact Windows and Asymmetric 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 unusual values lie mainly on one side of a distribution? The ordinary midmean keeps the middle half by position. The shortest-half midmean instead searches for the most compact half-window and averages that window, allowing the retained block to move away from asymmetric contamination.
The method connects sorting, sliding windows, range minimisation and averages. It is a specialised robust location estimator whose compactness rule must be stated clearly.
Did you know? The shortest half can sit left or right of the ordinary middle half. It follows the tightest cluster rather than assuming unusual values are balanced across both tails.
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
Symmetric trimming removes similar amounts from both ends. If contamination lies mainly in one tail, that fixed central window can still be pulled.
The shortest-half midmean searches consecutive ordered windows and selects the one with the smallest range.
NIST describes the estimator as the mean of the most compact half of the data. It calls this an asymmetric version of the midmean and notes lower efficiency than the median but reduced sensitivity to asymmetrically distributed outliers.
The exact number of retained observations depends on n and the documented convention.
| Data | Window size | Winning window |
|---|---|---|
| 1,2,3,4,5,20,21,100 | 5 | 1,2,3,4,5 |
Consider the ordered values 1, 2, 3, 4, 5, 20, 21 and 100. Under the NIST even-n convention, examine consecutive windows of five observations.
The first window 1,2,3,4,5 has range 4. The next windows have ranges 18, 18 and 96.
| Winning range | Winning sum | Shortest-half midmean |
|---|---|---|
| 4 | 15 | 3 |
The unique shortest range is 4, so retain 1,2,3,4,5.
Their mean is 15/5 = 3. The shortest-half midmean is therefore 3 under this stated convention.
The simple middle-half values are 3,4,5,20, whose mean is 8 under a four-position teaching convention. That central window still includes the start of the right-side gap.
The shortest-half rule moves left to the denser cluster.
The raw median is (4+5)/2 = 4.5. The shortest-half midmean is 3 because the compact block is asymmetric around the raw centre.
Neither value is automatically correct for every scientific question; they estimate different location features.
After sorting, list every consecutive window of the required size. For each, calculate maximum minus minimum.
Choose the minimum-range window and average its values. Preserve all window ranges for audit.
Two windows can have the same shortest range but different means. A procedure needs a declared tie rule, such as first window, last window or average across tied candidates.
Do not choose the tied window that gives the preferred conclusion.
If unusual values form their own very tight cluster and are numerous enough, the shortest window may select that cluster instead of the intended population.
Compactness is not proof of validity. Sampling and domain context remain essential.
NIST notes that the estimator has rather low efficiency, lower than the median, while offering resistance to asymmetric outliers.
That tradeoff shows why robustness must be matched to a plausible contamination pattern.
Adding a constant to every value adds it to every window mean while leaving ranges unchanged. Multiplying by a positive factor multiplies ranges and the final estimate by that factor without changing the selected window.
These properties are useful tests.
With few observations, moving one value can change which window is shortest. The estimator can jump rather than move smoothly.
Report n, retained count and selected indices.
Apply the missing-data rule before sorting. Keep original IDs so selected observations can be traced back.
Never lose provenance when the algorithm reorders data.
Write the eight values on cards and slide a frame covering five cards. Record each frame’s range and mean.
Students can see why the first frame wins and how right-tail contamination changes an ordinary central window.
Sort a copy, calculate each rolling window’s start, end, range and mean, then select the minimum range under the tie rule.
Highlight retained IDs and leave the raw order untouched.
Sort values with stable IDs, generate consecutive windows of the documented size, find the minimum range and apply a deterministic tie rule.
Unit-test the example to select 1–5 and return mean 3.
The shortest-half midmean does not identify errors, guarantee the main population or work well when contamination forms a compact rival cluster.
It is a location estimator, not a confidence interval or probability.
Compact-window estimators appear in robust interlaboratory analysis and exploratory measurement studies with asymmetric contamination. Similar window-search ideas occur in signal processing and clustering.
The transferable skill is to make an optimisation rule explicit.
The midmean guide uses the fixed middle half. The trimmed-mean guide removes declared tails, while shortest-half midmean searches for compactness.
Students should draw every candidate window and write the range before the mean. This separates the selection criterion from the reported centre.
Parents can ask what happens if two windows tie or if an unusual cluster is itself compact.
Find the shortest-half midmean for 2,3,4,5,20,21,22,100 and document ties. Translate all values by 10. Double all values and predict the selected window.
Construct a compact outlier cluster and explain the failure risk.
### Is it the same as the ordinary midmean? No. It searches for the most compact half rather than automatically using the middle positions.
### Why can the estimate differ from the median? The compact window can be asymmetric around the raw centre.
### What if two windows have equal range? Use and report a prespecified deterministic tie rule.
### Does the shortest window prove those values are correct? No. Compactness is a mathematical property, not a validity certificate.
Read the official NIST shortest-half midmean documentation, the NIST midmean discussion and the cited Duewer robust-location article record. The worked window search is original.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
The shortest-half midmean turns robust location into a visible search for compactness. It helps students understand both the power and the risk of following the tightest cluster.
Extended practice laboratory
Practice case 1: Sort the data
Setup: Use the article’s shortest-half 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: Set window size
Setup: Use the article’s shortest-half 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: List windows
Setup: Use the article’s shortest-half 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: Calculate ranges
Setup: Use the article’s shortest-half 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: Choose the minimum
Setup: Use the article’s shortest-half 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: Average selected window
Setup: Use the article’s shortest-half 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: Move right-tail value
Setup: Use the article’s shortest-half 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: Move central value
Setup: Use the article’s shortest-half 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: Create a range tie
Setup: Use the article’s shortest-half 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: Apply tie rule
Setup: Use the article’s shortest-half 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 midmean
Setup: Use the article’s shortest-half 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 median
Setup: Use the article’s shortest-half 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 trimmed mean
Setup: Use the article’s shortest-half 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: Translate all values
Setup: Use the article’s shortest-half 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: Scale all values
Setup: Use the article’s shortest-half 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: Create compact contamination
Setup: Use the article’s shortest-half 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: Use small n
Setup: Use the article’s shortest-half 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 IDs
Setup: Use the article’s shortest-half 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: Handle missing values
Setup: Use the article’s shortest-half 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: Plot windows
Setup: Use the article’s shortest-half 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: Audit efficiency
Setup: Use the article’s shortest-half 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 validity claims
Setup: Use the article’s shortest-half 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: Name the target
Setup: Use the article’s shortest-half 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 compact-window conclusion
Setup: Use the article’s shortest-half 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.