D'Agostino's K-squared test
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In statistics, D'Agostino's K2 test is a goodness-of-fit measure of departure from normality, based on transformations of the sample kurtosis and skewness. The test statistic K2 is obtained as follows:
In the following derivation, n is the number of observations (or degrees of freedom in general); 
is the sample skewness, b2 is the sample kurtosis, defined as
where μ3 and μ4 are the third and fourth central moments, respectively, 
is the sample mean, and σ2 is the second central moment, the variance.
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[edit] Transformed Skewness
First, calculate 
, a transformation of the skewness that is approximately normally distributed under the null hypothesis that the data are normally distributed.
[edit] Transformed Kurtosis
Next, calculate 
, a transformation of the kurtosis b2 that is approximately normally distributed under the null hypothesis that the data are normally distributed.
Next, compute the skewness of the kurtosis:
![A = 6 + \frac{8}{\sqrt{\beta_1\left(b_2\right)}} \left[ \frac{2}{\sqrt{\beta_1\left(b_2\right)}} + \sqrt{1+\frac{4}{\beta_1\left(b_2\right)}}\right]](../../../../math/4/6/8/46831e91bb65139a5339811d50e45edb.png)
![Z\left(b_2\right) = \left(\left(1 - \frac{2}{9A}\right) - \sqrt[3]{\frac{1-2/A}{1+x\sqrt{2/(A-4)}}}\right)\sqrt{\frac{9A}{2}}](../../../../math/d/b/3/db397980155b9d54b322f7678aa102e7.png)
[edit] Omnibus K2 statistic
Now, we can combine and to define D'Agostino's Ombibus K2 test for normality.
K2 is approximately distributed as χ2 with 2 degrees of freedom.
[edit] References
- D'Agostino, Ralph B., Albert Belanger, and Ralph B. D'Agostino, Jr. "A Suggestion for Using Powerful and Informative Tests of Normality", The American Statistician, Vol. 44, No. 4. (Nov., 1990), pp. 316-321.
