- The frequency test is a test of uniformity.
- Two different methods available, Kolmogorov-Smirnov test and the chi-square test. Both tests measure the agreement between the distribution of a sample of generated random numbers and the theoretical uniform distribution.
- Both tests are based on the null hypothesis of no significant difference between the sample distribution and the theoretical distribution.

**The Kolmogorov-Smirnov test**- This test compares the
cdf of uniform distribution
*F(x)*to the empirical cdf of the sample of*N*observations.- As
*N*becomes larger, should be close to*F(x)* - Kolmogorov-Smirnov test is based on the statistic

that is the absolute value of the differences. - Here
*D*is a random variable, its sampling distribution is tabulated in Table A.8. - If the calcualted
*D*value is greater than the ones listed in the Table, the hypothesis (no disagreement between the samples and the theoretical value) should be rejected; otherwise, we don't have enough information to reject it. - Following steps are taken to perform the test.
- Rank the data from smallest to largest

- Compute

- Compute
- Determine the critical value, , from
Table A.8 for the specified significance level and the given
sample size
*N*. - If the sample statistic
*D*is greater than the critical value , the null hypothsis that the sample data is from a uniform distribution is rejected; if , then there is no evidence to reject it.

- Rank the data from smallest to largest
- Example 8.6 on page 300.

**Chi-Square test**- The chi-square test looks at the
issue from the same angle but uses different method. Instead of
measure the difference of each point between the samples and the true
distribution, chi-square checks the ``deviation'' from the
``expected'' value.

where*n*is the number of classes (e.g. intervals), is the number of samples obseved in the interval, is expected number of samples in the interval. If the sample size is*N*, in a uniform distribution,

See Example 8.7 on page 302.