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. Shapiro and R. S. Francia in 1972 as a simplification of the Shapiro–Wilk test.

Theory

Let

x

(

i

)

{\displaystyle x_{(i)}}

be the

i

{\displaystyle i}

-th ordered value from our size-

n

{\displaystyle n}

sample. For example, if the sample consists of the values

{

5.6

,

−

1.2

,

7.8

,

3.4

}

{\displaystyle \left\{5.6,-1.2,7.8,3.4\right\}}

,

x

(

2

)

=

3.4

{\displaystyle x_{(2)}=3.4}

, because that is the second-lowest value. Let

m

i

:

n

{\displaystyle m_{i:n}}

be the mean of the

i

{\displaystyle i}

th order statistic when making

n

{\displaystyle n}

independent draws from a normal distribution. For example,

m

2

:

4

≈

−

0.297

{\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

x

{\displaystyle x}

and the

m

{\displaystyle m}

:

W

′

=

cov

⁡

(

x

,

m

)

σ

x

σ

m

=

∑

i

=

1

n

(

x

(

i

)

−

x

¯

)

(

m

i

−

m

¯

)

(

∑

i

=

1

n

(

x

(

i

)

−

x

¯

)

2

)

(

∑

i

=

1

n

(

m

i

−

m

¯

)

2

)

{\displaystyle W'={\frac {\operatorname {cov} (x,m)}{\sigma _{x}\sigma _{m}}}={\frac {\sum _{i=1}^{n}(x_{(i)}-{\bar {x}})(m_{i}-{\bar {m}})}{\sqrt {\left(\sum _{i=1}^{n}(x_{(i)}-{\bar {x}})^{2}\right)\left(\sum _{i=1}^{n}(m_{i}-{\bar {m}})^{2}\right)}}}}

Under the null hypothesis that the data is drawn from a normal distribution, this correlation will be strong, so

W

′

{\displaystyle W'}

values will cluster just under 1, with the peak becoming narrower and closer to 1 as

n

{\displaystyle n}

increases. If the data deviate strongly from a normal distribution,

W

′

{\displaystyle W'}

will be smaller.

This test is a formalization of the older practice of forming a Q–Q plot to compare two distributions, with the

x

{\displaystyle x}

playing the role of the quantile points of the sample distribution and the

m

{\displaystyle m}

playing the role of the corresponding quantile points of a normal distribution.

Compared to the Shapiro–Wilk test statistic

W

{\displaystyle W}

, the Shapiro–Francia test statistic

W

′

{\displaystyle W'}

is easier to compute, because it does not require that we form and invert the matrix of covariances between order statistics.