Summarize and model exact data

Keep every summary exact

The statistics plugin is written in RiX. Its linear quantile policy uses the exact rank p*(n-1), and both variance conventions remain rational.

Runnable RiXRun cell

.Plugin.Load("stats"); values := [1/3, 2/3, 5/3, 7/3]; summary := .stats.Summary(values); [summary[:mean], summary[:median], summary[:populationVariance]];

Analyze interval measurements without midpoint substitution

Generate intervals from stated centers and error bounds, summarize every admissible point dataset, and make a known-scale z decision over the whole measurement box.

Runnable RiXRun cell

.Plugin.Load("stats"); measured := .stats.MeasurementIntervals([10,12,14],[1,1/2,2]); summary := .stats.IntervalSummary(measured); test := .stats.IntervalOneSampleZTest(measured,10,2,:greater); decision := test.Decision(1/20,{= absoluteWidth=1/500,maxWork=400 }); [summary[:meanRange],summary[:minimumRange],summary[:maximumRange],decision[:status]];

When the population scale is estimated from the observations, use the interval t procedures. They enclose every test obtained by selecting one exact value from each measurement interval:

Runnable RiXRun cell

oneSample := .stats.IntervalOneSampleTTest( [99/10:101/10,117/10:123/10,139/10:141/10], 10, :greater, {= maxBoxes=128 } ); paired := .stats.IntervalPairedTTest( [99/10:101/10,119/10:121/10,139/10:141/10], [79/10:81/10,89/10:91/10,109/10:111/10], 0, :greater, {= maxBoxes=128 } ); welch := .stats.IntervalTwoSampleTTest( [9:10,10:11,11:12], [5:6,6:7,7:8], 0, {= alternative=:twoSided,maxBoxes=128 } ); [oneSample.PValueBounds(),paired.Decision()[:status],welch[:degreesOfFreedomRange]];

ANOVA, correlation, and simple-regression slope tests use the same certified measurement-box interpretation:

Runnable RiXRun cell

anova := .stats.IntervalOneWayANOVA([ [1:11/10,2:21/10,3:31/10], [4:41/10,5:51/10,6:61/10] ],{= maxBoxes=128 }); correlation := .stats.IntervalCorrelationTest( [1:11/10,2:21/10,3:31/10,4:41/10], [2:21/10,4:41/10,6:61/10,8:81/10], :greater, {= maxBoxes=128 } ); slope := .stats.IntervalRegressionSlopeTest( [1:11/10,2:21/10,3:31/10,4:41/10], [2:21/10,4:41/10,6:61/10,8:81/10], 0, :greater, {= maxBoxes=128 } ); [anova.PValueBounds(),correlation.PValueBounds(),slope.PValueBounds()];

Subdivision only tightens an enclosure. A larger maxBoxes costs more work; it cannot remove a true result. If a box still permits zero sample variance, constant predictors, or perfect fits, RiX keeps infinite statistic endpoints and widens the p-value bounds honestly.

Build portable representations

Tables and plots are core output values. The same histogram and box plot can therefore be displayed in RiX Web or exported through any compatible renderer.

Runnable RiXRun cell

.Plugin.Load("stats"); values := [1, 2, 2, 3, 5, 8]; .Fragment([ .stats.SummaryTable(values), .Figure(.stats.HistogramGraphic(values, {= bins=3 }), "Three exact bins"), .Figure(.stats.BoxPlot(values), "Linear-interpolation quartiles") ]);

Variance is the population statistic. Use SampleVariance when the values are a sample; it reports a clear error for fewer than two observations.

Use the compatibility normal helpers

The distribution functions return refinable reals. NormalQuantile is kept distinct from the exact sample Quantile operation.

First-class probability laws and simulation live in .probability; these scalar helpers remain available for existing statistics worksheets.

Runnable RiXRun cell

.Plugin.Load("stats"); values := [ .stats.NormalPDF(0), .stats.NormalCDF(1), .stats.NormalQuantile(975/1000) ]; values.Map((value) -> .numerics.Refine(value, {= absoluteWidth=1/1000, maxWork=12000 }));

Pass optional mean and exact positive Rational standard deviation arguments for another normal distribution, for example .stats.NormalCDF(12,10,2).

Confidence records

Confidence procedures return semantic records whose endpoints remain certified expressions. The method name states which normal-theory formula was used.

Runnable RiXRun cell

.Plugin.Load("stats"); meanInterval := .stats.MeanConfidence([3,4,5,6,7], 95/100); proportionInterval := .stats.ProportionConfidence(17, 25, 95/100); [meanInterval, proportionInterval];

Test means and proportions

Test results keep exact sample bookkeeping and certified reference-function values. The method and assumptions travel with the answer.

Runnable RiXRun cell

.Plugin.Load("stats"); knownScale := .stats.OneSampleZTest([48,49,50,52,54],50,2,:greater); student := .stats.OneSampleTTest([2,4,6,8,10],0,:twoSided); proportion := .stats.OneProportionZTest(60,100,1/2,:greater); .Fragment([ knownScale.Table(), student.Table(), proportion.Table() ]);

Use :less, :greater, or :twoSided to state the alternative. A decision refines the p-value before comparing it with alpha:

Runnable RiXRun cell

decision := student.Decision(1/20,{= absoluteWidth=1/10000,maxWork=2000 }); [decision[:status],decision[:pValueInterval],decision[:conclusion]];

The status is :reject, :failToReject, or :unresolved. The last case is important: it means the certified p-value enclosure still crosses alpha and more refinement is needed, not that the null hypothesis was accepted.

For two independent samples the default is Welch's unequal-variance formula. The current certified Student-t tail kernel has integer degrees of freedom, so a fractional Welch-Satterthwaite value is retained while its floor is exposed as referenceDegreesOfFreedom for the p-value approximation.

Runnable RiXRun cell

welch := .stats.TwoSampleTTest([1,2,3,4],[6,8,10,12]); pooled := .stats.TwoSampleTTest([8,9,10,11],[3,4,5,6],0,{= equalVariance=1 }); paired := .stats.PairedTTest([10,12,13,15],[8,9,11,12]); [ welch[:degreesOfFreedom],welch[:referenceDegreesOfFreedom],welch[:pValueStatus], pooled[:method],paired[:differences] ];

Compare several groups and categorical counts

One-way ANOVA reports exact sums and mean squares before its certified F tail. Pearson goodness-of-fit and independence tests retain their expected counts and cell contributions for inspection.

Runnable RiXRun cell

.Plugin.Load("stats"); anova := .stats.OneWayANOVA([[1,2,3],[4,5,6],[3,4,5]]); goodness := .stats.ChiSquareGoodnessOfFit([20,30,50],[25,25,50]); independence := .stats.ChiSquareIndependence([[12,8,10],[6,14,10]]); .Fragment([anova.Table(),goodness.Table(),independence.Table()]);

Read the assumptions field before interpreting a p-value. In particular, these procedures do not infer that observations were independently sampled or that normality, equal variance, and expected-count conditions hold.

Exact simple linear regression

Coefficients and residuals are exact for exact inputs. Only diagnostics that require square roots become certified refinable values.

Runnable RiXRun cell

.Plugin.Load("stats"); model := .stats.LinearRegression([1,2,3,4], [3,5,7,9]); .Fragment([ .stats.RegressionTable(model), .stats.ResidualTable(model), .Paragraph(@"The fitted value at x=5 is @{.stats.Predict(model,5)}.") ]);

Probability simulations retain their provenance when summarized.

Runnable RiXRun cell

.Plugin.Load("probability"); .Plugin.Load("stats"); run := .probability.Dice(2,6).Simulate(20, {= seed=42 }); summary := .stats.SimulationSummary(run); theory := .stats.DistributionSummary(.probability.Dice(2,6)); [summary[:mean], summary[:sourceFamily], summary[:sourceSeed], summary[:sourceSamplingPolicy], theory[:mean], theory[:variance]];

Reuse a portable color scale

Runnable RiXRun cell

.Plugin.Load("plot"); .Plugin.Load("stats"); scale := .plot.ColorScale({= colors=["#172554","#38bdf8","#f8fafc"],minimum=0,maximum=6 }); .stats.HistogramGraphic([1,2,2,3,4,4,4],{= bins=3,colorScale=scale });

The histogram retains the complete rix.color-scale@1 value, including its exact bounds, so another plot family can apply the same policy.