The Kolmogorov Smrinov’s one sample test is a test for goodness of fit. The Kolmogorov Smrinov’s one sample test is concerned with the degree of agreement between the distribution of the observed sample values and some specified theoretical distribution. It determines whether or not the values in a sample can reasonably be thought to have come from a population having a theoretical distribution.
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In Kolmogorov Smrinov’s one sample test, it is assumed that the distribution of the underlying variables being tested is continuous in nature. It is appropriate for those types of variables that are tested at least on an ordinal scale. One usually conducts the test in order to test the normality assumption in analysis of variances.
Suppose, for example, F0(x) has a completely specified cumulative relative frequency distribution function in Kolmogorov Smrinov’s one sample test. In this case the theoretical distribution under the null hypothesis for any value of F0(x) is the proportion of the cases that are expected to have values which are equal to or are less than the value of x.
Suppose Sn(x) is the observed cumulative relative frequency distribution function of a random sample of ‘n’ observations in Kolmogorov Smrinov’s one sample test. If xi is any possible value then Sn(xi) = Fi/n , where Fi is nothing but the number of expected proportions of observations which are less than or equal to xi.
Now, according to the null hypothesis in Kolmogorov Smrinov’s one sample test, it is expected that for every value of xi, Sn(xi) should be fairly close to F0(xi). In other words, if the null hypothesis is true, then the difference between Sn(xi) and F0(xi) is small and should be within the limits of the random error.
The Kolmogorov Smrinov’s one sample test focuses on the largest of the deviations. The largest deviation is called the maximum deviation. The maximum deviation is the largest absolute difference between the cumulative observed proportion and the cumulative proportion expected on the basis of the hypothesized distribution. The sampling distribution of the maximum deviation under the null hypothesis is generally known.
There are certain assumptions that are made in Kolmogorov Smrinov’s one sample test.
It is assumed that the sample is drawn from the population by the process of random sampling.
It is assumed that the level of data variables should be continuous interval or ratio types in order to get the exact results. If approximate results are required by the researcher through Kolmogorov Smrinov’s one sample test, then the researcher can use ordinal data or grouped interval level of data.
Kolmogorov Smrinov’s one sample test is also used for ordinal scale of data when the large-sample assumptions of the chi-square goodness-of-fit test are not met.
The hypothetical distribution is specified in advance in Kolmogorov Smrinov’s one sample test.
In the case of the normal distribution the expected sample mean and sample standard deviation should always be specified in advance.
In the case of Poisson distribution and in the case of exponential distribution in Kolmogorov Smrinov’s one sample test, the expected sample mean should always be specified in advance.
In the case of uniform distribution in Kolmogorov Smrinov’s one sample test, the expected range which consists of the minimum and maximum values, should always be specified in advance.