Shapiro–Francia test

The Shapiro–Francia test is a statistical test for the normality of a population, based on sample data. It was introduced by S. S. Shaprio and R. S. Francia in 1972 as a simplification of the Shapiro–Wilk test.

Let ${\displaystyle x_{i}}$ be the ${\displaystyle i}$th ordered value from our size-${\displaystyle n}$ sample. For example, if the sample consists of the values ${\displaystyle \left\{5.6,-1.2,7.8,3.4\right\}}$, ${\displaystyle x_{2}=3.4}$, because that is the second-lowest value. Let ${\displaystyle m_{i:n}}$ be the mean of the ${\displaystyle i}$th order statistic when making ${\displaystyle n}$ independent draws from a normal distribution. For example ${\displaystyle m_{2:4}\approx -0.297}$, meaning that the second-lowest value in a sample of four draws from a normal distribution is typically about 0.297 standard deviations below the mean. Form the Pearson correlation coefficient between the ${\displaystyle x}$ and the ${\displaystyle m}$:

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