Home / Shared Math Details / Shared Estimators and Tests / Kolmogorov-Smirnov Test
Kolmogorov-Smirnov Test¶
This page defines the Kolmogorov-Smirnov normality test, with the Lilliefors correction, 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 |
| \(s\) | sample standard deviation (\(n - 1\) denominator) |
| \(F_n\) | empirical cumulative distribution function |
| \(\Phi\) | standard normal cumulative distribution function |
| \(z_{(i)}\) | standardized sorted value, \((x_{(i)} - \bar{x})/s\) |
| \(D\) | Kolmogorov-Smirnov statistic |
Test statistic¶
\[ D^{+} = \max_i\left(F_n(x_{(i)}) - \Phi(z_{(i)})\right), \qquad
D^{-} = \max_i\left(\Phi(z_{(i)}) - F_n(x_{(i-1)})\right), \qquad
D = \max(D^{+}, D^{-}) \]
\(F_n\) is the empirical cumulative distribution function of the sorted data, and \(z_{(i)} = (x_{(i)} - \bar{x})/s\) standardizes each value with the sample mean and sample standard deviation. \(D\) is the largest vertical distance between the empirical CDF and the fitted normal CDF.
p-value¶
Because the normal parameters are estimated from the same data, the p-value uses the Lilliefors correction rather than the classical Kolmogorov-Smirnov distribution.
Used by¶
- Summary Statistics: the normality-test section.
- Cpk/Histogram Math Details: the selectable goodness-of-fit method.
See Also¶
References¶
- Massey, F. J. (1951). The Kolmogorov-Smirnov test for goodness of fit. Journal of the American Statistical Association, 46(253), 68-78.
- Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(318), 399-402.
- D'Agostino, R. B., and Stephens, M. A. (1986). Goodness-of-Fit Techniques. Marcel Dekker.