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eduKateSG · Why Mathematics? · Mountain Plots, Folded Percentiles and Difference Distributions
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 paired differences are skewed, heavy-tailed or easier to compare through percentiles than a normal model? A mountain plot folds an empirical cumulative distribution at 50%. The peak marks the median, the horizontal position shows bias and the sides reveal spread, tails, gaps and outliers.
This chapter uses nine original differences: −4, −2, −1, 0, 1, 1, 2, 3 and 5. With percentile positions rank divided by n + 1, the unfolded levels are 10% to 90%. Folding values above 50% creates heights 10, 20, 30, 40, 50, 40, 30, 20 and 10. The peak sits at difference one, making the sample median visible without assuming normality.
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
Calculate Y − X for every paired item and sort the results. Our ordered values are already −4, −2, −1, 0, 1, 1, 2, 3 and 5.
A mountain plot is a graph of this empirical distribution, not a new measurement model. Its meaning depends on correct pairing, shared units and a declared sign convention.
The same differences used in a Bland–Altman plot can be viewed through this percentile lens.
| Difference | Rank percentile | Folded height |
|---|---|---|
| -4 | 10% | 10% |
| -2 | 20% | 20% |
| -1 | 30% | 30% |
| 0 | 40% | 40% |
| 1 | 50% | 50% |
| 1 | 60% | 40% |
| 2 | 70% | 30% |
| 3 | 80% | 20% |
| 5 | 90% | 10% |
There are n = 9 observations. Use the plotting position 100r/(n + 1) for rank r.
Ranks one through nine therefore receive 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80% and 90%.
Other plotting-position formulas exist. Report the convention because tail coordinates can differ in small samples.
Keep percentile values up to 50 unchanged. For values above 50, replace p by 100 − p.
The heights become 10, 20, 30, 40, 50, 40, 30, 20 and 10.
Plot each height against its corresponding difference. The result rises to a central peak and falls, resembling a mountain.
The 50% point corresponds to the fifth ordered difference, which is one. The peak is therefore centred at x = 1.
If zero represents perfect method agreement, a peak at one suggests a positive median bias in this sample.
The plot shows location visually but does not by itself provide a confidence interval for the median.
The left tail reaches −4 and the right tail reaches 5 at the 10% plotting level. The right side extends farther, hinting at positive-tail asymmetry.
Long tails represent large differences. Gaps in the x coordinates show regions with no observed differences.
With nine observations, tail impressions are unstable. The graph is exploratory evidence, not a population guarantee.
Did you know that mathematical work on folded cumulative distributions links the area under a folded curve with mean absolute deviation from a quantile?
This connects a visual shape with a numerical measure of spread. The graph is not merely decorative.
The exact relationship depends on the continuous distribution representation and chosen fold, so avoid estimating population quantities by eye from a tiny sample.
Mountain plots are especially useful when two or three new methods are each compared with the same reference. Overlay their folded difference distributions.
Peaks show median shifts, widths show central spread and tails show extremes. Use distinct accessible line styles, not colour alone.
All curves must use the same difference definition, units, plotting positions and axis scale.
A Bland–Altman plot preserves the relationship between difference and measurement magnitude. A mountain plot focuses on the marginal distribution of differences.
The mountain can make percentiles and skewness easier to compare. It cannot reveal proportional bias across magnitude because the pair mean is absent.
Use both when range-dependent disagreement matters.
Horizontal levels correspond to tail probabilities. A 2.5% fold level can mark endpoints related to the central 95% empirical range when sample size is sufficient.
With only nine observations, such an extreme level is not supported by observed ranks. Pretending otherwise would interpolate beyond meaningful tail information.
Match sample size to the percentiles you want to discuss.
Our data contain two differences equal to one. Empirical distributions are step functions, and ties create vertical or overlapping plot behaviour depending on implementation.
Do not add random noise simply to make a smoother mountain. Ties may reflect measurement resolution.
State how software assigns ranks to ties and whether smoothing was applied.
A smoothed curve may be easier to view, but it can round away gaps, ties or small secondary features.
Show points or the unsmoothed empirical curve when the sample is modest. If smoothing is used, name the method and parameter.
The data, not the aesthetic curve, remain the evidence.
The x axis may show absolute differences or percentage differences. The choice changes the practical question.
Percentage calculations need a clear denominator and caution near zero. Ratios can be preferable for positive values.
Label the axis with units or percent and describe the reference method.
The differences −4 and 5 sit in the outer tails. They influence the horizontal span even though the peak remains one.
Investigate their source rather than deleting them because they make the mountain wide.
An outlier may reveal a range limit, data error or important failure mode.
With n observations, plotting positions are discrete. A small n produces a coarse mountain and weak tail resolution.
Larger samples create more percentile levels but do not fix biased sampling. Representative range and conditions still matter.
Report n directly on or beside the plot.
Two methods can have the same difference distribution while one fails only at high values and the other fails randomly.
The mountain cannot distinguish those patterns. Check differences against pair means or reference values.
Every visualisation removes information; good analysis combines views.
The mountain plot is primarily exploratory. Bootstrap bands for quantile curves may be useful if resampling respects clusters and paired units.
Do not treat visual overlap as a formal equivalence test. Predefined tolerances and probability criteria are needed for an agreement decision.
A graph guides questions; it does not replace design-based inference.
Define reference and comparison methods, units and difference direction. Calculate paired differences and order them.
Choose and report plotting positions, fold the percentiles and draw the unsmoothed points or curve. Compare peaks, widths and tails.
Add Bland–Altman or regression views for magnitude patterns and use tolerances for decisions.
Write ranks one to nine, divide each by ten and convert to percentages. Fold values above 50 by subtracting from 100.
Pair the heights with ordered differences and sketch the mountain. Mark zero and the peak at one.
Then add a new large difference of 20. Recalculate plotting positions with n = 10 and observe how both ranks and tail span change.
- The peak marks a median, not a mean.
- A narrow mountain does not prove absence of systematic bias.
- The plot does not show magnitude-dependent error.
- Visual overlap is not proof of equivalence.
- Smoothing does not create information.
- Tail percentiles require enough observations.
The plot is most useful when its construction is transparent.
Using the nine differences:
- verify ranks and 10% plotting increments;
- reproduce all folded heights;
- identify the median difference one;
- compare left and right tail reach;
- find the proportion at or below zero; and
- sketch a second mountain after adding two to every difference.
Explain why shifting every difference moves the mountain horizontally without changing its shape.
Students can build a mountain plot from differences between two timers or repeated estimates. It turns ordering and percentiles into a visible object.
Parents can ask what the peak and tails mean, and what information the graph leaves out.
This supports statistics, graph literacy and critical thinking for science, technology and AI data.
Why fold at 50%?
Folding places both tails below a central median peak, making distribution comparison compact.
Is a mountain plot nonparametric?
Its empirical construction does not require a normal distribution, though inference or smoothing may add assumptions.
Can it replace a Bland–Altman plot?
No. It complements difference-versus-magnitude information rather than preserving it.
What does a peak away from zero mean?
It indicates a nonzero sample median difference under the declared subtraction direction.
Monti’s primary paper is Folded Empirical Distribution Function Curves—Mountain Plots, *The American Statistician* 49(4), 1995. The mountainplot package introduction documents distribution features the plot can reveal.
Continue with Bland–Altman limits and the Mathematics Learning Hub.
Verify pairing, difference direction, units, sorted values, plotting-position formula, tie handling, smoothing, sample size and reference method. Add a magnitude plot and tolerance criterion before making interchangeability claims.
The importance of mathematics is that a list of differences becomes a compact picture of median, spread and tails without hiding how the picture was built.
Extended practice laboratory
Practice case 1: Median peak
Setup: Use nine ordered differences. Task: Locate 50%.
Check: The peak sits at difference one. 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: Fold 60%
Setup: Transform the percentile. Task: Calculate height.
Check: One hundred minus sixty gives forty. 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: Fold 90%
Setup: Transform again. Task: Calculate height.
Check: The tail point has height ten. 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: Shift all data
Setup: Add two to each difference. Task: Predict the curve.
Check: Shape stays fixed and moves right. 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: Scale all data
Setup: Multiply differences by ten. Task: Predict the curve.
Check: Horizontal spread scales while percentile heights stay fixed. 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: Reverse sign
Setup: Multiply by minus one. Task: Describe geometry.
Check: The mountain mirrors horizontally after reordering. 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: Add an outlier
Setup: Append difference twenty. Task: Recompute n.
Check: Ranks, plotting positions and tail span change. 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: Duplicate median
Setup: Add another one. Task: Handle ties.
Check: The empirical step around the peak becomes wider. 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: Normal-looking data
Setup: Use symmetric differences. Task: Read shape.
Check: A centred symmetric mountain has balanced sides. 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: Skewed data
Setup: Stretch the right tail. Task: Read shape.
Check: The right base extends farther than the left. 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: Two methods
Setup: Overlay two mountains. Task: Set controls.
Check: Use common units, reference and plotting positions. 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: Different sample sizes
Setup: Overlay n nine and n ninety. Task: Avoid confusion.
Check: Line resolution differs and n must be shown. 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: Smoothing
Setup: Apply a curve. Task: Protect evidence.
Check: Keep points visible and disclose the smoother. 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: Gap
Setup: Remove middle differences. Task: Inspect shape.
Check: Horizontal gaps reveal missing regions in the distribution. 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: Tail percentile
Setup: Seek 2.5% with n nine. Task: Assess support.
Check: The sample is too small for a stable empirical endpoint. 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: Absolute differences
Setup: Fold unsigned values. Task: Notice the loss.
Check: Direction and median bias disappear. 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: Percentage differences
Setup: Normalise by reference. Task: Check zeros.
Check: The denominator and near-zero behaviour must be documented. 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: Magnitude bias
Setup: Errors grow at high values. Task: Name the blind spot.
Check: Mountain plots omit pair magnitude. 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: Bland comparison
Setup: Use the same differences. Task: Combine views.
Check: One shows distribution; the other preserves magnitude position. 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: Bootstrap band
Setup: Resample observations. Task: Respect design.
Check: Clustered pairs require cluster resampling. 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: Visual overlap
Setup: Two curves overlap. Task: Avoid equivalence claim.
Check: A graph is not a formal probability criterion. 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: Wide tail
Setup: A few extremes appear. Task: Investigate.
Check: Do not delete valid failures to narrow the mountain. 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: Axis truncation
Setup: Crop the horizontal range. Task: Identify distortion.
Check: Hidden tails create a misleadingly compact plot. 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: Audit file
Setup: Recreate the plot. Task: Save essentials.
Check: Preserve differences, ranks, plotting formula, ties, smoother and axis units. 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.