# Statistics - Kolmogorov Smirnov Test

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:

## Formula

$D = Maximum |F_o(X)-F_r(X)|$

Where −

${F_o(X)}$ = Observed cumulative frequency distribution of a random sample of n observations.

and ${F_o(X) = \frac{k}{n}}$ = (No.of observations ≤ X)/(Total no.of observations).

${F_r(X)}$ = The theoretical frequency distribution.

The critical value of ${D}$ 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.

### Example

**Problem Statement:**

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.

B.Sc. | B.A. | B.Com | M.A. | M.Com | |
---|---|---|---|---|---|

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.

**Solution:**

${H_o}$: 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 | ${F_O(X)}$ | ${F_T(X)}$ | ${|F_O(X)-F_T(X)|}$ | |
---|---|---|---|---|---|

Observed (O) | Theoretical (T) | ||||

B.Sc. | 5 | 12 | 5/60 | 12/60 | 7/60 |

B.A. | 9 | 12 | 14/60 | 24/60 | 10/60 |

B.COM. | 11 | 12 | 25/60 | 36/60 | 11/60 |

M.A. | 16 | 12 | 41/60 | 48/60 | 7/60 |

M.COM. | 19 | 12 | 60/40 | 60/60 | 60/60 |

Total | n=60 | ||||

Test statistic ${|D|}$ 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.

## Formula

${D = Maximum |{F_n}_1(X)-{F_n}_2(X)|}$

Where −

${n_1}$ = Observations from first sample.

${n_2}$ = Observations from second sample.

It has been seen that when the cumulative distributions show large maximum deviation ${|D|}$ it is indicating towards a difference between the two sample distributions.

The critical value of D for samples where ${n_1 = n_2}$ and is ≤ 40, the K-S table for two sample case is used. When ${n_1}$ and/or ${n_2}$ > 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.