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eduKateSG · Why Mathematics? · Midmean, the Middle Half and Stable 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 19–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when we want the average of the central half of a distribution? The midmean, also called an interquartile mean in many contexts, averages observations between the 25th and 75th percentiles. It ignores the outer quarters and describes the spacing inside the middle half.
The method connects quartiles, ordering and arithmetic mean. It is a fixed central-fraction summary, distinct from a median and from a general trimmed mean whose tail percentages may vary.
Did you know? The midmean can use several central observations even when the raw median depends on only one or two positions.
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 19–25 · Study and verify
The mean responds to every value, including extremes. The median responds mainly to the central order position.
The midmean averages the middle 50%, limiting tails while retaining spacing inside that region.
NIST defines the midmean as the mean of observations between the 25th and 75th percentiles. It lists the midmean among alternative location measures that are less affected by tail extremes.
Exact boundary handling depends on the percentile convention. That convention must be stated for small samples.
| Original | Middle half | Midmean |
|---|---|---|
| 1,2,3,4,5,6,7,100 | 3,4,5,6 | 4.5 |
Consider 1, 2, 3, 4, 5, 6, 7 and 100. Under the simple order-position convention used here, the middle four observations are 3, 4, 5 and 6.
Their sum is 18, so the midmean is 18/4 = 4.5.
| Mean | Median | Midmean |
|---|---|---|
| 16 | 4.5 | 4.5 |
The ordinary mean is 128/8 = 16. The median is (4 + 5)/2 = 4.5. The midmean is also 4.5 in this symmetric central example.
Equality here is a property of the example, not a general identity.
The value 100 lies above the upper-quartile region and does not enter this midmean. Changing it to 1,000 leaves the result unchanged under the same order positions.
The raw value remains scientifically important and should still be shown.
The midmean is essentially a fixed middle-half mean. A general trimmed mean can retain 80%, 90% or another declared central fraction.
The fixed definition gives midmean a clear intent but does not make it universally preferable.
The median uses the centre order position. The midmean averages a wider central region and responds to changes among those retained observations.
It can therefore describe central spacing more fully while remaining resistant to outer quarters.
The ordinary mean has important algebraic and inferential properties. The midmean deliberately changes the target by excluding tail quarters.
Always label it. Calling it simply “the average” hides the design.
Quantile algorithms may interpolate boundaries or choose different included positions. With small n, these choices can noticeably change the midmean.
Report the software and percentile type or list retained observations directly.
If many values equal a quartile boundary, an index-based rule may include some tied records and exclude others. The numerical result may still be reproducible, but its explanation needs care.
Preserve IDs and show the rule.
The middle half of a skewed distribution may not be symmetric around its median. Midmean can shift toward the longer side within the interquartile region.
Plot the data and interpret the estimator as central-half location rather than a universal centre.
Adding a constant to every observation adds it to the midmean. Multiplying all observations by a positive factor multiplies the midmean by that factor.
These checks help verify unit conversions and formulas.
Small samples leave few observations in the central half. One change can therefore move the result sharply.
Large samples make percentile boundaries more stable, but sampling design remains decisive.
Place eight ordered values on a number line, shade the outer two on each side and average the four central values. Then move 100 farther right.
Next change 6 to 20 and observe that a central-half change does affect the midmean.
Sort a copy, calculate quartile boundaries under a declared convention, mark included rows and average their values.
Keep original row IDs and record boundary treatment.
Specify the quantile algorithm, determine the retained central region and return the midmean together with included count and boundary values.
Unit-test the eight-value example to obtain 4.5 under the stated order-position rule.
A point estimate does not supply its own standard error. Bootstrap or analytic inference requires a sampling plan and a method compatible with the estimator.
Do not reuse ordinary-mean uncertainty formulas automatically.
Midmean does not make data normal, detect outliers or repair non-representative sampling. It can conceal important tails if reported alone.
It is one view of location, not a verdict on which observations matter.
Central-fraction averages appear in exploratory statistics, finance, quality studies and comparisons where tail resistance is useful. The idea also helps students understand quartiles as operational boundaries.
The transferable skill is to connect a verbal definition with exact included positions.
The Mathematics Learning Hub connects this fixed central-half summary with general trimmed means, while the Winsorised-mean guide keeps tail positions through replacement. The Hodges–Lehmann guide builds location from pairwise averages.
Students should draw the ordered positions and circle the values being averaged. This makes quartile conventions visible.
Parents can ask whether the central-half summary is being used alongside a plot that preserves the tails.
Find the midmean of 2, 3, 4, 5, 6, 7, 8 and 80. Change the top value and verify resistance. Change one central value and explain the response.
Compare mean, median and midmean for a skewed dataset.
### Is the midmean the midpoint of the quartiles? No. It is the mean of observations in the interquartile region under the stated convention.
### Is it always equal to the median? No. The worked equality is specific to the example.
### Are tails deleted permanently? No. They are excluded from this summary and should remain in the raw data.
### Why can software disagree? Different percentile and boundary conventions can retain different values.
Read the official NIST measures-of-location handbook, the NIST difference-of-midmeans documentation and the NIST trimmed-mean documentation. The worked example states its boundary convention explicitly.
Explore the Mathematics Learning Hub and How Mathematics Works for related foundations.
The midmean turns quartiles into an averaging rule. It shows students that robust location is built by choosing which part of a distribution the summary is meant to represent.
Extended practice laboratory
Practice case 1: Sort eight values
Setup: Use the article’s midmean 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: Locate the middle half
Setup: Use the article’s midmean 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: Average retained values
Setup: Use the article’s midmean 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: Move the far value
Setup: Use the article’s midmean 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 a central value
Setup: Use the article’s midmean 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: Compare mean
Setup: Use the article’s midmean 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: Compare median
Setup: Use the article’s midmean 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: Compare trimmed mean
Setup: Use the article’s midmean example and change one declared input for case 8. Task: Recalculate from raw values, preserve units and state which mathematical or design condition changed.
Check: Check the relevant invariant, separate the number from its claim, and name the next diagnostic or source of evidence. Rebuild the relevant counts before interpreting the coefficient, preserve the stated design, and write one sentence separating the numerical result from the claim it can support. Then vary one defensible assumption, calculate again, and explain whether the conclusion changes because of new data, a new reference model, or a new decision purpose. Finish by naming the unit, tolerance, and evidence needed for transfer. Record one graph or table that would expose failure hidden by the summary.
Practice case 9: Change quantile type
Setup: Use the article’s midmean 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: Handle ties
Setup: Use the article’s midmean 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: Inspect skewness
Setup: Use the article’s midmean 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: Translate all data
Setup: Use the article’s midmean 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: Scale all data
Setup: Use the article’s midmean 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: Use small n
Setup: Use the article’s midmean 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: Use large n
Setup: Use the article’s midmean 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: Preserve row IDs
Setup: Use the article’s midmean 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 values
Setup: Use the article’s midmean 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: Record boundaries
Setup: Use the article’s midmean 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 all values
Setup: Use the article’s midmean 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: Estimate uncertainty
Setup: Use the article’s midmean 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: Avoid normality claims
Setup: Use the article’s midmean 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: Explain central-half target
Setup: Use the article’s midmean 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: Check software output
Setup: Use the article’s midmean 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 midmean conclusion
Setup: Use the article’s midmean 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.