Probability, statistics, and data analysis with RiX

An accessible, runnable course guide from finite probability through inference, regression, simulation, and interval-valued measurements.

This is a stand-alone guide for a first undergraduate course in probability, statistics, or data analysis. You can read it in order or start from the part that matches your course. Every RiX block marked Run is checked by the RiX documentation build and can be copied into the REPL, RiX Web, or a notebook.

RiX keeps three jobs separate:

• .probability describes what could happen: laws, events, exact probabilities, and simulations;
• .data stores and reshapes observations in typed tables; and
• .stats summarizes observations, fits simple models, and performs inference.

Load only the plugins needed by a calculation. RiX uses exact integers and rationals by default. A value such as 10:11 is a closed rational interval, not the two-point array [10,11] and not a floating-point approximation.

1. A ten-minute start

Exact probability

The probability of exactly five heads in ten fair tosses is a rational number.

.Plugin.Load("probability");
coins := .probability.Binomial(10,1/2);
[coins.PMF(5),coins.CDF(5),coins.Mean(),coins.Variance()];

Output

[63/256, 319/512, 5, 2..1/2]

Describe observed data

Summary computes exact descriptive statistics. Input order does not matter.

.Plugin.Load("stats");
summary := .stats.Summary([8,5,9,7,6]);
[summary[:minimum],summary[:median],summary[:mean],summary[:sampleVariance]];

Output

[5, 7, 7, 2..1/2]

Keep measurement uncertainty

Suppose three distances were recorded only to within stated error bounds. The mean is then a range containing the mean of every point dataset consistent with the measurements.

.Plugin.Load("stats");
distances := [99/10:101/10,119/10:121/10,139/10:141/10];
summary := .stats.IntervalSummary(distances);
[summary[:meanRange],summary[:minimumRange],summary[:maximumRange]];

Output

[11..9/10:12..1/10, 9..9/10:10..1/10, 13..9/10:14..1/10]

RiX does not replace those intervals by their midpoints unless you explicitly ask for the midpoint summary. That prevents a neat-looking point answer from claiming more than the measurements establish.

2. Finite probability

Counting

.Plugin.Load("probability");
[
    .probability.Choose(52,5),
    .probability.Permutations(10,3),
    .probability.MultinomialCoefficient([2,3,1])
];

Output

[2598960, 720, 60]

Use bounded enumeration for genuinely small sample spaces. Here the 36 ordered two-die outcomes make the event “sum equals seven” transparent.

.Plugin.Load("probability");
rolls := .probability.CartesianPower([1,2,3,4,5,6],2);
seven := .probability.Event(rolls,(roll)->roll[1]+roll[2]==7);
[seven[:favorableCount],seven[:outcomeCount],seven[:probability]];

Output

[6, 36, 1/6]

For larger repeated experiments, use a direct distribution such as Binomial, Multinomial, Dice, or CardDraw rather than materializing the whole sample space.

Conditional probability and Bayes’ rule

Conditional returns another finite distribution. This example conditions a fair die on an even result.

.Plugin.Load("probability");
die := .probability.Finite([1,2,3,4,5,6],[1,1,1,1,1,1]);
even := .probability.Conditional(die,(x)->x%2==0);
[
    even.PMF(4),
    .probability.ConditionalProbability(die,(x)->x>=5,(x)->x%2==0)
];

Output

[1/3, 1/3]

Bayes(prior,likelihood under H,likelihood under not-H) exposes every term of the update. In a screening example with a 1% prior prevalence, 99% sensitivity, and 1% false-positive rate, a positive result gives a posterior of one half.

.Plugin.Load("probability");
update := .probability.Bayes(1/100,99/100,1/100);
[update[:prior],update[:evidence],update[:posterior]];

Output

[1/100, 99/5000, 1/2]

Random variables, joint laws, and moments

A random variable is a function of an outcome. RandomVariable pushes a finite law through such a function; duplicate transformed outcomes have their probabilities combined by the resulting PMF.

.Plugin.Load("probability");
die := .probability.Finite([1,2,3,4,5,6],[1,1,1,1,1,1]);
parity := .probability.RandomVariable(die,(x)->x%2);
joint := .probability.Joint(die,(x)->x%2,(x)->x>=4);
[
    parity.PMF(0),
    .probability.Marginal(joint,2).PMF(1),
    .probability.ExpectedValue(die),
    .probability.VarianceOf(die),
    .probability.Covariance(die,(x)->x,(x)->2*x)
];

Output

[1/2, 1/2, 3..1/2, 2..11/12, 5..5/6]

Independent(joint,predicate1,predicate2) returns 1 or 0 by checking the exact product rule. These finite-law tools are excellent for tables, dice, cards, and classroom Bayes exercises. Sums/convolutions of arbitrary user laws and mixtures remain a later probability slice.

3. Standard distributions and simulation

The Phase 1–2 library includes Bernoulli, binomial, multinomial, categorical, geometric, negative-binomial, hypergeometric, Poisson, uniform, exponential, gamma, beta, chi-square, normal, Student-t, F, lognormal, and Cauchy laws. Discrete laws use PMF; continuous laws use PDF. Both use CDF, Quantile, Mean, Variance, Sample, and Simulate where mathematically applicable.

.Plugin.Load("probability");
geometric := .probability.Geometric(1/3);
normal := .probability.Normal(10,2);
cauchy := .probability.Cauchy();
[
    geometric.PMF(2),geometric.CDF(4),
    .numerics.Refine(normal.CDF(10),{= absoluteWidth=1/1000,maxWork=1000 })[:interval],
    cauchy.Mean(),cauchy.Variance(),cauchy.MomentsExist()
];

Output

[4/27, 211/243, 1/2:1/2, _, _, 0]

A simulation record keeps the seed and sampling policy. Repeating the seed replays the experiment.

.Plugin.Load("probability");
dice := .probability.Dice(2,6);
first := dice.Simulate(12,{= seed=2026 });
second := dice.Simulate(12,{= seed=2026 });
[first[:values]==second[:values],first[:samplingPolicy],first[:exactSampling]];

Output

[_, exactUniformDice, 1]

Finite rational sampling is exact. Continuous simulation uses a disclosed finite inverse-CDF grid, so its record says exactSampling=0. This is useful simulation, but it is not mislabeled as an exact draw from a continuum.

4. Typed data tables

Create a relation from a schema and rows. Types are Any, Integer, Rational, Number, Interval, and String; columns are nullable unless nullable=0 is stated.

.Plugin.Load("data");
observations := .data.Relation([
    {= id="student",type=:String,nullable=0 },
    {= id="section",type=:String,nullable=0 },
    {= id="score",type=:Rational }
],[
    ["Ada","A",9/10],
    ["Bo","B",4/5],
    ["Cy","A",_],
    ["Di","B",7/10]
]);
known := .data.Missing(observations,["score"],:drop);
ordered := .data.Sort(known,["score"],{= descending=1 });
.data.TableView(ordered,{= caption="Known scores" });

Output

Known scores
student  section  score
-------  -------  -----
    Ada        A   9/10
     Bo        B    4/5
     Di        B   7/10

Useful cleaning and reshaping operations include Project, Rename, Distinct, Filter, Sort, Join, Calculate, and Missing. Grouped aggregation keeps rational sums and means exact.

.Plugin.Load("data");
scores := .data.Relation([
    {= id="section",type=:String,nullable=0 },
    {= id="score",type=:Rational,nullable=0 }
],[["A",4/5],["A",9/10],["B",7/10],["B",3/4]]);
groups := .data.Group(scores,["section"]);
summary := .data.Aggregate(groups,[
    {= id="n",op=:count },
    {= id="mean",column="score",op=:mean },
    {= id="minimum",column="score",op=:min }
]);
.data.Rows(summary);

Output

[{= section=A, n=2, mean=17/20, minimum=4/5 }, {= section=B, n=2, mean=29/40, minimum=7/10 }]

Frequency produces tidy counts and exact proportions. Contingency produces the count matrix and margins expected by a chi-square independence test.

.Plugin.Load("data");
.Plugin.Load("stats");
survey := .data.Relation([
    {= id="group",type=:String,nullable=0 },
    {= id="answer",type=:String,nullable=0 }
],[["A","yes"],["A","yes"],["A","no"],["B","yes"],["B","no"],["B","no"]]);
frequencies := .data.Frequency(survey,["group","answer"]);
contingency := .data.Contingency(survey,"group","answer");
test := .stats.ChiSquareIndependence(contingency[:counts]);
[.data.Rows(frequencies),contingency[:rowTotals],test[:statistic],test.Decision()[:status]];

Output

[[{= group=A, answer=yes, count=2, proportion=1/3 }, {= group=A, answer=no, count=1, proportion=1/6 }, {= group=B, answer=yes, count=1, proportion=1/6 }, {= group=B, answer=no, count=2, proportion=1/3 }], [3, 3], 2/3, failToReject]

5. Descriptive statistics and graphics

Summary, SummaryTable, Histogram, HistogramGraphic, and BoxPlot cover the common first-course descriptive workflow. Quantiles use exact linear interpolation at rank p*(n-1).

.Plugin.Load("stats");
values := [8,2,5,5,9,4,7];
summary := .stats.Summary(values);
histogram := .stats.Histogram(values,4);
[
    .stats.SummaryTable(values),
    .stats.HistogramGraphic(histogram),
    .stats.BoxPlot(values)
];

Output

[Exact descriptive statistics
          statistic     value
-------------------  --------
              count         7
            minimum         2
                 q1    4..1/2
             median         5
                 q3    7..1/2
            maximum         9
               mean    5..5/7
population variance   5..3/49
    sample variance  5..19/21, [Graphic: 420 × 220, 5 scene nodes], [Graphic: 420 × 120, 5 scene nodes]]

The graphics use exact coordinates. Renderers lower them outward-safely, so a displayed exact bar, box, or endpoint encloses what RiX knows rather than shaving a boundary through ordinary floating rounding.

6. Confidence intervals and tests

RiX records the method, reference distribution, assumptions, and p-value status alongside the headline result. It does not claim to have verified that observations are independent or normally distributed.

Common procedures are:

• one- and two-sample z tests with known population scales;
• one- and two-proportion score z tests;
• one-sample, paired, pooled, and Welch t tests;
• one-way ANOVA;
• Pearson chi-square goodness-of-fit and independence tests;
• Pearson correlation and its t test; and
• simple linear regression and its slope t test.

.Plugin.Load("stats");
tTest := .stats.OneSampleTTest([12,11,15,14,13],10,:greater);
anova := .stats.OneWayANOVA([[8,9,10],[11,12,13],[9,10,11]]);
goodness := .stats.ChiSquareGoodnessOfFit([20,30,50],[25,25,50]);
[
    tTest[:estimate],tTest[:degreesOfFreedom],tTest.Decision()[:status],
    anova[:statistic],anova.Decision()[:status],
    goodness[:statistic],goodness.Decision()[:status]
];

Output

[13, 4, reject, 7, reject, 2, failToReject]

For a confidence interval using Student’s t reference, use MeanTConfidence; related operations cover paired and independent mean differences. The critical-value search returns an outward bracket and uses its upper endpoint, making the reported interval conservatively wide.

.Plugin.Load("stats");
confidence := .stats.MeanTConfidence([1,2,3,4,5],9/10,{=
    extent=10,
    criticalWidth=1/100,
    probabilityWidth=1/10000,
    maxWork=600,
    maxIterations=16
});
[
    confidence[:estimate],confidence[:degreesOfFreedom],
    confidence[:criticalValueInterval],confidence[:criticalPolicy]
];

Output

[3, 4, 2..33/256:2..71/512, outwardUpper]

TestDecision(alpha) refines a certified p-value. It returns :unresolved instead of guessing if the enclosure still crosses alpha. “Fail to reject” is not evidence that the null hypothesis is true.

7. Correlation and regression

.Plugin.Load("stats");
x := [1,2,3,4,5];
y := [2,3,5,4,7];
model := .stats.LinearRegression(x,y);
slopeTest := .stats.RegressionSlopeTest(model);
[
    .stats.Correlation(x,y),
    model[:intercept],model[:slope],model[:rSquared],
    .stats.Predict(model,6),slopeTest[:test]
];

Output

[{= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=div, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=11 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=148 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=148 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=148 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=148 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=div, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=11 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=148 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=148 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=148 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=148 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=div } }, 9/10, 1..1/10, 121/148, 7..1/2, regressionSlopeT]

RegressionTable(model) and ResidualTable(model) make the fit inspectable. The usual linear-model assumptions still belong in the analysis; a small p-value does not repair dependence, curvature, changing variance, or biased sampling.

8. Interval-valued measurements

Enter measured values directly

Use closed intervals for rounded distances, clock readings, weights, or counts reported with a known tolerance. An Interval data column also accepts an exact point, treating it as a zero-width measurement.

.Plugin.Load("data");
loads := .data.Relation([
    {= id="batch",type=:String,nullable=0 },
    {= id="weight",type=:Interval,nullable=0 }
],[
    ["A",995/10:1005/10],
    ["A",1000/10:1010/10],
    ["A",99:100]
]);
summary := .data.Aggregate(.data.Group(loads,["batch"]),[
    {= id="total",column="weight",op=:sum },
    {= id="mean",column="weight",op=:mean },
    {= id="minimum",column="weight",op=:min },
    {= id="maximum",column="weight",op=:max }
]);
.data.Rows(summary);

Output

[{= batch=A, total=298..1/2:301..1/2, mean=99..1/2:100..1/2, minimum=99:100, maximum=100:101 }]

For interval inputs, sum and mean use interval arithmetic. The minimum range is [minimum of lows, minimum of highs]; the maximum range is [maximum of lows, maximum of highs]. These are the tight ranges obtainable when each row may be any point in its stated interval.

Generate interval data for an exploration

MeasurementIntervals(centers,errors) creates [center-error,center+error] for each row. errors may be one nonnegative exact value or an array. Here the latent readings and resolution classes are both replayable simulated data.

.Plugin.Load("probability");
.Plugin.Load("stats");
latent := .probability.Dice(2,6).Simulate(12,{= seed=41 })[:values];
errors := .probability.Finite([1/10,1/5,1/2],[3,2,1]).Simulate(12,{= seed=42 })[:values];
measured := .stats.MeasurementIntervals(latent,errors);
summary := .stats.IntervalSummary(measured);
[measured,summary[:meanRange],summary[:maximumMeasurementWidth]];

Output

[[2..9/10:3..1/10, 7..9/10:8..1/10, 8..1/2:9..1/2, 8..9/10:9..1/10, 5..1/2:6..1/2, 3..4/5:4..1/5, 5..4/5:6..1/5, 8..4/5:9..1/5, 7..9/10:8..1/10, 8..9/10:9..1/10, 6..9/10:7..1/10, 10..9/10:11..1/10], 7..9/40:7..73/120, 1]

This pattern lets a class vary instrument resolution, rounding policy, or counting tolerance independently of the latent process. For genuinely simulated measurement error rather than bounded ignorance, simulate an error variable and then separately record the interval guaranteed by the instrument. Those are different models and should remain visibly different.

Test every point dataset allowed by the measurements

IntervalOneSampleZTest covers the transparent case of a known population standard deviation. It finds the exact range of possible sample means and, by monotonicity of the normal tail, the smallest and largest possible p-values over the whole measurement box.

.Plugin.Load("stats");
measurements := [9:11,23/2:25/2,12:16];
test := .stats.IntervalOneSampleZTest(measurements,10,2,:greater);
decision := test.Decision(1/20,{= absoluteWidth=1/500,maxWork=400 });
[
    test[:measurementMeanRange],test[:effectRange],
    test[:pValueLower],test[:pValueUpper],decision[:status]
];

Output

[10..5/6:13..1/6, 5/6:3..1/6, {= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=7..25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=7..25/48 } } } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=sub } }, {= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=normalCDF, input={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, evidencelevel=proof, provenance={= plugin=numerics, version=8, algorithm=directNormalRationalBounds, function=normalCDF, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=25/48 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=25/48 } } } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=sub } }, measurementDependent]

When the scale must be estimated, the interval t, ANOVA, correlation, and regression procedures use exact interval arithmetic with bounded subdivision. Each reported range encloses every result obtained by choosing one point from each measurement interval. Subdivision reduces repeated-variable overestimation; it never removes an admissible result.

.Plugin.Load("stats");
one := .stats.IntervalOneSampleTTest(
    [9:10,11:12,13:14],10,:greater,{= maxBoxes=16 }
);
anova := .stats.IntervalOneWayANOVA([
    [1:11/10,2:21/10,3:31/10],
    [4:41/10,5:51/10,6:61/10]
],{= maxBoxes=16 });
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=16 }
);
[
    one[:statisticEndpoints],one.PValueBounds(),
    anova.PValueBounds(),correlation.PValueBounds()
];

Output

[[{= sign=1, squared=864/1777, infinite=0 }, {= sign=1, squared=5..479/485, infinite=0 }], [{= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=mul } }, parameters={= operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1452/1937 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=1452/1937 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=mul } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=sub } }, {= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=432/2209 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=432/2209 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=432/2209 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=432/2209 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } }, parameters={= operation=mul, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=adapter, constructor=refinableAdapter, procedure=delegatedRefinement, source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=432/2209 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=432/2209 } }, parameters={= source={= valuekind=numericsAlgorithmReal, schema=rix.numerics.algorithm-real@1, kind=weightedNthRoot, degree=2, radicand={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=432/2209 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, evidencelevel=proof, provenance={= plugin=numerics, version=2, algorithm=weightedNthRoot, source=432/2209 } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=certifiedSingletonAdapter, backend=numerics } } }, declaredproperties=[range, existence, separation, consistency, singularity, closure], provenance={= plugin=oracle, version=2, source=arithmeticRecipe, operation=mul } }, parameters={= operation=sub, left={= valuekind=oracle, schema=rix.oracle@1, kind=complete, constructor=rational, procedure=singular, parameters={= value=1/2 }, eta=_, declaredproperties=[range, existence, separation, disjointness, consistency, singularity, closure], choicepolicy=single, seed=1, provenance={= plugin=oracle, version=1, source=rationalConstructor } }, right={= valuekind=oracle, schema=rix.oracle@1, kind=arithmetic, constructor=arithmetic, procedure=intervalRefinement, 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The remaining entry points are IntervalPairedTTest, IntervalTwoSampleTTest (Welch by default, pooled with equalVariance=1), and IntervalRegressionSlopeTest. Increase maxBoxes when dependency inflation is too wide, or set measurementWidth to stop once individual boxes reach a chosen coordinate width. If a denominator can be zero, RiX reports an infinite statistic endpoint and safely widens the p-value range.

The status has a deliberately stronger vocabulary than an ordinary point-data test:

• :rejectForAllMeasurements: every admissible point dataset rejects;
• :failToRejectForAllMeasurements: none rejects;
• :measurementDependent: some admissible datasets reject and some do not;
• :unresolved: refine the certified p-value bounds further.

Measurement uncertainty and sampling uncertainty are not interchangeable. The interval z procedure varies the unknown measured points while using the specified population scale for sampling variability. The interval t and F procedures vary both the estimate and its data-derived standard error. Using midpoints with an ordinary test would answer a different question.

9. Explore the Central Limit Theorem—and its boundary

For iid variables with finite mean and positive finite variance, standardized sums approach a normal law. Exact dice sums make this visible without Monte Carlo error; uniform and exponential simulations give continuous examples.

The Cauchy distribution is the essential counterexample: it has no mean or variance, and the average of independent standard Cauchy variables is again standard Cauchy. More averaging does not concentrate it around zero.

.Plugin.Load("probability");
dice := [1,2,4,8].Map((n)->.probability.Dice(n,6));
uniformRun := .probability.Uniform(-1,1).Simulate(8,{= seed=12,grid=128 });
cauchyRun := .probability.Cauchy().Simulate(8,{= seed=12,grid=128 });
[
    dice.Map((law)->[law.Mean(),law.Variance()]),
    uniformRun[:samplingPolicy],cauchyRun[:samplingPolicy],
    .probability.Cauchy().MomentsExist()
];

Output

[[[3..1/2, 2..11/12], [7, 5..5/6], [14, 11..2/3], [28, 23..1/3]], finiteUniformGrid, finiteInverseCDFGrid, 0]

Continue with the repository’s explanatory exploration, explorations/probability/central-limit-theorem.md, and its runnable companion central-limit-theorem.rix. It compares exact dice convolutions with uniform, exponential, normal, and Cauchy behavior while keeping simulation policies visible.

10. A practical analysis checklist

1. State the population, sampling process, variables, and units.
2. Put rows and types into a .data.Relation; make missing-data policy explicit.
3. Preserve bounded measurement uncertainty with Interval columns.
4. Plot and summarize before choosing a test.
5. Choose the procedure from the design and assumptions, not from the desired p-value.
6. Inspect the result’s method, assumptions, degrees of freedom, and p-value status.
7. Report an effect estimate and confidence interval, not only a decision.
8. Separate sampling uncertainty, bounded measurement uncertainty, and simulation approximation in the write-up.
9. Keep the seed and samplingPolicy when reporting simulations.
10. If a certified comparison is unresolved, request more refinement or report that it remains unresolved.

That separation is the central advantage of RiX for these courses: exact finite calculations remain exact, certified real calculations remain enclosures, interval measurements remain intervals, and simulations retain their approximation provenance.