Shapiro-Wilk Test vs Kolmogorov-Smirnov Test

 Shapiro-Wilk Test vs Kolmogorov-Smirnov Test


 



Shapiro-Wilk Test

Purpose: Tests whether a sample comes from a normal distribution.

 Methodology: Compares the sample distribution to a reference normal distribution using a linear combination of order statistics.

 Assumptions:

     Observations are independent.

     Data is continuous.

 Outputs:

     W-statistic: A measure of the difference between the sample and normal distributions.

     P-value: The probability of obtaining the observed W-statistic or more extreme if the data is truly normal.

 

Advantages:

     More sensitive to departures from normality than other tests.

     Can detect non-linearities and skewness.

 Disadvantages:

     Not as powerful as the Kolmogorov-Smirnov test for extreme deviations from normality.

     Computationally more demanding.


Kolmogorov-Smirnov Test


 Purpose: Tests whether a sample comes from a specified distribution (not necessarily normal).

 Methodology: Compares the cumulative distribution function (CDF) of the sample to the CDF of the specified distribution.

 Assumptions:

     Observations are independent.

     Data is continuous.

     Specified distribution is known.

 Outputs:

     D-statistic: The maximum distance between the sample CDF and the specified distribution CDF.

     P-value: The probability of obtaining the observed D-statistic or more extreme if the data is truly from the specified distribution.

 Advantages:

     Can test for any specified distribution.

     Powerful for detecting large deviations from the specified distribution.


 Disadvantages:

     Less sensitive to non-linearity and skewness than the Shapiro-Wilk test.

     Less powerful for small deviations from the specified distribution.


Comparison


When to Use Each Test:


Shapiro-Wilk Test: Tests the suitability of sample data with a normal distribution.

Kolmogorov-Smirnov Test: Tests the suitability of sample data with a specific distribution.

Statistical Basis:

Shapiro-Wilk Test: Uses the W test statistic.

Kolmogorov-Smirnov Test: Uses the D test statistic.

Sensitivity to Sample Size:

Shapiro-Wilk Test: More sensitive to small sample sizes (n < 50).

Kolmogorov-Smirnov Test: More suitable for larger sample sizes.




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