This test is used in situations where a comparison has to be made between an observed sample distribution and theoretical distribution.
K-S One Sample Test
This test is used as a test of goodness of fit and is ideal when the size of the sample is small. It compares the cumulative distribution function for a variable with a specified distribution. The null hypothesis assumes no difference between the observed and theoretical distribution and the value of test statistic ‘D’ is calculated as:
· Fo(X)Fo(X) = Observed cumulative frequency distribution of a random sample of n observations.
· and Fo(X)=knFo(X)=kn = (No.of observations ≤ X)/(Total no.of observations).
· Fr(X)Fr(X) = The theoretical frequency distribution.
The critical value of DD is found from the K-S table values for one sample test.
Acceptance Criteria: If calculated value is less than critical value accept null hypothesis.
Rejection Criteria: If calculated value is greater than table value reject null hypothesis.
In a study done from various streams of a college 60 students, with equal number of students drawn from each stream, are we interviewed and their intention to join the Drama Club of college was noted.
|No. in each class||5||9||11||16||19|
It was expected that 12 students from each class would join the Drama Club. Using the K-S test to find if there is any difference among student classes with regard to their intention of joining the Drama Club.
HoHo: There is no difference among students of different streams with respect to their intention of joining the drama club.
We develop the cumulative frequencies for observed and theoretical distributions.
|Streams||No. of students interested in joining||FO(X)FO(X)||FT(X)FT(X)|||FO(X)−FT(X)||FO(X)−FT(X)||
Test statistic |D||D| is calculated as:
The table value of D at 5% significance level is given by
Since the calculated value is greater than the critical value, hence we reject the null hypothesis and conclude that there is a difference among students of different streams in their intention of joining the Club.
K-S Two Sample Test
When instead of one, there are two independent samples then K-S two sample test can be used to test the agreement between two cumulative distributions. The null hypothesis states that there is no difference between the two distributions. The D-statistic is calculated in the same manner as the K-S One Sample Test.
· n1n1 = Observations from first sample.
· n2n2 = Observations from second sample.
It has been seen that when the cumulative distributions show large maximum deviation |D||D| it is indicating towards a difference between the two sample distributions.
The critical value of D for samples where n1=n2n1=n2 and is ≤ 40, the K-S table for two sample case is used. When n1n1 and/or n2n2 > 40 then the K-S table for large samples of two sample test should be used. The null hypothesis is accepted if the calculated value is less than the table value and vice-versa.
Thus use of any of these nonparametric tests helps a researcher to test the significance of his results when the characteristics of the target population are unknown or no assumptions had been made about them.