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Shapiro-Wilk Test

This page defines the Shapiro-Wilk normality test that Quantum XL uses. It is one of the selectable goodness-of-fit methods wherever a normality check is offered.

Notation

Term Description
\(n\) number of data values
\(x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}\) the data sorted in ascending order
\(\bar{x}\) sample mean
\(a_i\) Shapiro-Wilk coefficients from normal order statistics
\(W\) Shapiro-Wilk statistic

Test statistic

\[ W = \frac{\left(\sum_{i=1}^{n} a_i\, x_{(i)}\right)^2}{\sum_{i=1}^{n} (x_i - \bar{x})^2} \]

\(x_{(1)} \le \cdots \le x_{(n)}\) are the values sorted ascending and \(\bar{x}\) is their mean. The constants \(a_i\) are derived from the expected values and covariances of standard normal order statistics; Quantum XL computes them with Royston's Algorithm AS R94. \(W\) is close to \(1\) when the data follow a normal distribution.

p-value

The p-value comes from Royston's normalizing transformation. For \(n = 3\) the exact value \(p = \frac{6}{\pi}\left(\arcsin\sqrt{W} - \frac{\pi}{3}\right)\) is used. For \(4 \le n \le 11\) the statistic is transformed with \(y = -\ln\!\big(\gamma - \ln(1 - W)\big)\), and for \(n > 11\) with \(y = \ln(1 - W)\); in both cases \(y\) is standardized with sample-size-dependent mean and standard deviation polynomials, and the p-value is the upper tail of the standard normal distribution for the resulting \(z\).

Used by

See Also

References

  1. Shapiro, S. S., and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611.
  2. Royston, P. (1995). Remark AS R94: A remark on Algorithm AS 181: The W-test for normality. Journal of the Royal Statistical Society: Series C (Applied Statistics), 44(4), 547-551.
  3. Royston, P. (1992). Approximating the Shapiro-Wilk W-test for non-normality. Statistics and Computing, 2(3), 117-119.