KS-Test and its significance

We would have seen, Z-Test, t-test etc, that presumes that the distribution should be a Normal Distribution. Have we ever wondered, is there a test to tell if a if a particular sample is coming from a certain distribution. Kolomogrov-Smrinov Test or KS-Test comes to our rescue to clear this doubt.

Kolomogrov-Smrinov Test or KS-Test in short is a statistical test used to check the if a particular sample is coming from a certain distribution.

Suppose we have observations x1, x2, …xn that we think come from a distribution D. The KS test is used to evaluate:

• Null Hypothesis: The samples come from D
• Alternative Hypothesis: The samples do not come from D

We compare the cdf and edf of sample and compare them. If the difference is below a critical value, we say they are from same distribution. Following are the steps used for testing using KS-Test.

• Sort the observations in ascending order
• Calculate the EDF of sorted observations
• Calculate the CDF of observations
• Compute the absolute differences
• Record the maximum difference
• Calculate the critical value
• Reject or accept null hypothesis

EDF — Emprical Distribution Function of a series is calculated as shown below:

For a series of 17,21,23 etc., emperical df is computed as 1/12, 2/12,3/12 etc.,

We compare this EDF with CDF of X1 series by taking an absolute difference as below. We assume that X1 is from normal distribution with mean, 36.25 and std_dev 12.09151.

We take the maximum absolute difference between the values (0.58998)and to determine if this maximum value is statistically significant, we must refer to a Kolmogorov-Smirnov Table of critical values and find the number equal to n = 12 and α = .05.

The critical value turns out to be 0.3754.

Since our value 0.5899 > critical value, hence we reject the null hypothesis and we conclude that, the given series is not coming from Normal Distribution.

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