5 Conclusions

In this paper, we define and introduce a statistical method, called the $1\sigma$ distribution function deviation test, to test assumed distributions of sources. An example of application of the new test is also given. The method is based on the fact that, for an assumed distribution of sources, the variance of the distribution function of samples is known. It is verified that, when the random variables vary continuously, a sample passing the test also passes any other statistical test which depends on the deviation of a statistical function from its expected value at the $1\sigma$ confidence level. Specifically, in these cases, a sample passing

the test must also pass any mean-value test within the same confidence level. In addition, the sample also passes the Kolmogorov-Smirnov test at the same confidence level. Since the mean value method and the K-S test are usually applied in the study of the distribution of sources in astronomy, we expect that this method will be applicable in testing the distribution of gamma-ray bursts as well as the luminosity function of quasars.