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eduKateSG · Why Mathematics? · Shortest-Half Midrange, Compact Intervals and Robust Centres
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 19–27
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 few observations stretch far to one side of a dataset? The shortest-half midrange sorts the values, finds the narrowest interval containing the required half-sample, and reports the midpoint of that interval’s endpoints.
The mechanism joins order statistics, sliding windows, range and midpoint geometry. It is easy to draw on a number line and powerful for teaching why a compact cluster can describe location differently from the full range.
Did you know? The method uses only two endpoints after choosing the winning window. Most of its reasoning happens in the selection of the interval, not in the final average.
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 19–27 · Study and verify
The ordinary midrange averages the minimum and maximum, so one distant observation can move it dramatically. The shortest-half version first searches for a compact half-sample.
Its purpose is a resistant location summary, especially when contamination is asymmetric.
NIST describes ordered values and chooses the consecutive half-sample window with minimum width x_{k+m}−x_k. The estimate is (x_k+x_{k+m})/2 for that selected window.
The retained-count convention depends on n. State it explicitly rather than guessing from the phrase “half”.
| Data | Retained count | Winning interval |
|---|---|---|
| 1,2,3,4,5,20,21,100 | 5 | 1 to 5 |
Sort 1, 2, 3, 4, 5, 20, 21 and 100. Under the documented even-n convention used here, examine five-value windows.
Their endpoint widths are 4, 18, 18 and 96. The winning interval runs from 1 to 5, so the shortest-half midrange is (1+5)/2=3.
| Lower endpoint | Upper endpoint | Shortest-half midrange |
|---|---|---|
| 1 | 5 | (1+5)/2 = 3 |
Keep observations consecutive in sorted order. A window that skips an interior value while including a farther value cannot be the shortest interval for the same count.
List every eligible start position to avoid selecting by eye.
For each window, subtract its lower endpoint from its upper endpoint. Choose the minimum width first.
Only then average the winning endpoints. Averaging all observations in the window would produce the shortest-half midmean, a related but distinct estimator.
The midpoint is the point equally far from the interval’s two boundaries. It describes the centre of the selected interval, not necessarily the balance point of its observations.
If values crowd near one endpoint, the midrange ignores that internal density.
Moving 100 farther right does not change the winning interval or estimate in the worked example. The distant value remains outside the compact half.
That illustrates resistance, but it does not guarantee immunity under every contamination pattern.
Both methods select the same kind of compact window. The midmean averages every retained value, while the midrange averages only the endpoints.
The shortest-half-midmean guide explains how internal values then affect the result.
The median depends on central order positions and needs no width search. It usually has better efficiency than the shortest-half midrange under common smooth models.
The compact-window estimator can be attractive when asymmetric outliers make a dense cluster the target.
For the worked data, the full midrange is (1+100)/2=50.5, while the shortest-half midrange is 3.
The difference comes from which endpoints are admitted, not from a different midpoint formula.
Two candidate intervals can have equal minimum width. A software package or analysis plan needs a deterministic tie rule.
Possible rules include the leftmost interval, the rightmost interval or an average of tied estimates. State the choice because results can differ.
If two dense clusters exist, the narrowest half may select one cluster and ignore another. That may be informative or misleading, depending on the target.
Plot the data and ask whether a single location summary is appropriate at all.
Adding a constant moves every endpoint and the estimate by that constant without changing widths. Multiplying by a positive factor multiplies widths and the estimate by that factor.
These are useful checks for a correct implementation.
With few observations, changing one value can alter the selected window abruptly. Odd sample sizes also require careful use of the documented retained-count rule.
Write the count m and the number of endpoints included before calculating.
CHAPTER 15 OF 27 · Explore applications
15. Missing values and observation identity
Remove or impute missing values only under a declared rule, then recompute n and the window count. Preserve observation IDs after sorting.
IDs let an analyst see whether the winning cluster comes from one batch, time or subgroup.
Place the eight worked values on a line. Slide a bracket covering five consecutive points and record its width at each position.
Students can then move 100, 21 or 5 and predict whether selection changes.
Sort values, create lower and upper endpoint columns, subtract to obtain widths, and use the minimum-width row to fetch the endpoints.
Add a count of minimum-width ties and display the selected IDs.
Sort value–ID pairs, generate every consecutive window of the required size, compute widths and apply a documented tie rule.
Unit-test the worked example for widths 4,18,18,96 and estimate 3.
The estimate can jump when two windows have nearly equal widths. Resampling may therefore produce an irregular distribution.
Uncertainty analysis should preserve dependence and the full selection algorithm rather than treating chosen endpoints as fixed in advance.
The shortest interval is not automatically the “true” group. The method does not prove that excluded values are errors.
It also does not measure spread; it is a location estimator constructed through a compactness rule.
Compact-window reasoning appears in robust statistics, metrology comparisons, clustering and signal processing. The transferable skill is separating a selection rule from the summary calculated after selection.
Students learn to audit discontinuous decisions rather than trusting a final number alone.
The midmean guide uses a fixed central half. The Hodges–Lehmann guide builds location from pairwise averages instead.
Students should show the sorted data, retained count, all candidate widths, tie rule and final midpoint.
Parents can ask whether a dot plot shows one cluster or several and whether excluded values were investigated.
Calculate the estimator for 0,1,2,3,10,11,12,100 under the same convention. Create a tie between two minimum-width windows and compare tie rules.
Translate every value by 50 and verify the predicted change.
### Is it the median of the shortest half? No. It is the midpoint of the selected interval’s endpoints.
### Does it always select the middle observations? No. The narrowest window may lie to one side.
### Are excluded values outliers? Not necessarily. Exclusion follows a compactness rule, not an error diagnosis.
### Why can the estimate jump? The winning window can switch when widths cross.
Read the official NIST shortest-half-midrange documentation and the companion NIST shortest-half-midmean documentation. The worked window widths and midpoint were independently recalculated.
Continue through the Mathematics Learning Hub and How Mathematics Works.
Shortest-half midrange turns location into a search problem: find the most compact eligible interval, then take its centre. That two-stage logic makes its strengths and limits visible.
Extended practice laboratory
Practice case 1: Sort the data
Setup: Use the article’s shortest-half-midrange 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 the retained count
Setup: Use the article’s shortest-half-midrange 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 candidate intervals
Setup: Use the article’s shortest-half-midrange 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 interval widths
Setup: Use the article’s shortest-half-midrange 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 shortest interval
Setup: Use the article’s shortest-half-midrange 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 its endpoints
Setup: Use the article’s shortest-half-midrange 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 the right-tail value
Setup: Use the article’s shortest-half-midrange 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 a central value
Setup: Use the article’s shortest-half-midrange 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 width tie
Setup: Use the article’s shortest-half-midrange 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 a tie rule
Setup: Use the article’s shortest-half-midrange 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 shortest-half midmean
Setup: Use the article’s shortest-half-midrange 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-midrange 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 full midrange
Setup: Use the article’s shortest-half-midrange 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 every value
Setup: Use the article’s shortest-half-midrange 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 every value
Setup: Use the article’s shortest-half-midrange 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 odd n
Setup: Use the article’s shortest-half-midrange 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-midrange 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 observation IDs
Setup: Use the article’s shortest-half-midrange 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-midrange 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 candidate intervals
Setup: Use the article’s shortest-half-midrange 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-midrange 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 multimodality
Setup: Use the article’s shortest-half-midrange 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 representativeness claims
Setup: Use the article’s shortest-half-midrange 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-interval conclusion
Setup: Use the article’s shortest-half-midrange 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.