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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, at least 3 values; 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 values or its range is below 1e-19, a constant sample included, the two cases Royston's AS R94 refuses (its first and sixth faults). AS R94 divides by the range throughout; here the sums run on values divided by the range past a range of 1e100, where a square would overflow, W being unchanged by the scale. A NaN or infinite range answers NaN for both, a case the algorithm leaves undefined (decision 0010). ArgumentOutOfRangeException when an option is outside its enumeration.

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-value floor is Royston's, and so is the 5000-value warning. AS R94 refuses fewer than three values, and past 5000 computes with a warning that its fitted transform is extrapolated; this does the same, without the warning:

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 algorithm needs at least 3 values…

scipy answers NaN for two values where the algorithm refuses them, and computes past 5000 as the algorithm does.

Omission runs before this test's own requirements, which are unchanged by it — a sample left too short, or left with a range below 1e-19, is refused exactly as it would be if passed directly.

Applies to — net10.0, netstandard2.0.

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