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Tests whether a sample could have come from a normal distribution.

public static TestResult Test(ReadOnlySpan<double> sample, NanPolicy nanPolicy = NanPolicy.Propagate)

Parameters — sample is the data, between 3 and 5000 values, not all equal; the span is read, never modified. nanPolicy says what to do with a NaN; scipy's nan_policy, defaulting to NanPolicy.Propagate.

Returns — TestResult: Royston's W statistic, and its p-value.

Exceptions — ArgumentException when sample holds fewer than 3 or more than 5000 values, or every value is identical.

Example — a symmetric sample against a heavily right-skewed one of the same size.

using Lodestar.Stats;

double[] symmetric = [-1.62, -1.10, -0.74, -0.47, -0.23, 0.0, 0.23, 0.47, 0.74, 1.10, 1.62];
double[] skewed = [0.1, 0.2, 0.3, 0.4, 0.6, 0.9, 1.4, 2.2, 3.6, 6.1, 12.0];

TestResult normalLooking = ShapiroWilk.Test(symmetric);
TestResult notNormal = ShapiroWilk.Test(skewed);

double symmetricW = Math.Round(normalLooking.Statistic, 4);   // => 0.9958
double skewedW = Math.Round(notNormal.Statistic, 4);          // => 0.7088
double skewedP = Math.Round(notNormal.PValue, 6);             // => 0.000608

Remarks — W is close to 1 for a shape close to normal, and the p-value follows from how far below 1 it falls. symmetric's W of 0.9958 carries a p-value over 0.999 — nothing here argues against normality; skewed's 0.7088 is a p-value of 0.0006, decisively against it.

The 3-to-5000 bound is a property of the fitted transform, not an arbitrary guard. Royston's polynomials that turn W into a p-value are fitted over that range; outside it there is no p-value the fit actually covers, so this refuses instead of extrapolating a number that would look precise and mean nothing:

using Lodestar.Stats;

string message = "nothing was thrown";
try
{
    ShapiroWilk.Test([1.0, 2.0]);
}
catch (ArgumentException error)
{
    message = error.Message;
}

string what = message;   // => Royston's approximation covers 3 to 5000 values…

scipy warns on the same input and answers anyway; this package treats the fitted range as a hard boundary instead.

Omission runs before this test's own requirements, which are unchanged by it — a sample left too short, or a pool left fully tied, is refused exactly as it would be if passed directly.

Applies to — net10.0, netstandard2.0.

See also — KolmogorovSmirnov.TwoSample, the Python equivalence table.