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The sampling in statistical surveys and numerical calculations as well as simulation testing of probabilistic models in virtually all fields of knowledge require a com-puter endowed with pseudorandom numbers generators. The main goal of the study is to compare the normal random number generators using various criteria. The properties of 12 random number generators for a normal distribution were investigated. Then, the family of generators was extended by two so-called application generators and a new approach for checking the quality of generators was adopted. A ready-made tool pre-pared in C++ and in Visual Basic for Application (VBA) for conducting self-contained research using generators was presented. All Monte Carlo simulations were carried out in C++, while the calculations were performed in the VBA editor using the Microsoft Excel 2016 spreadsheet. The analysis of the obtained results shows that the generators with best properties are: MP Monty Python, R, Biegun and Ziggurat. The worst generators, are: BM Box-Muller, Wallace, Iloraz and Excel.

The aim of the paper is to propose a new method of creating a Q-Q plot using the density function of order statistics as well as to compare it with the classical methods.The most popular distributions for statisticians are those flexible ones which have easily estimated parameters. In the pre-computer era Quantile-Quantile plot (Q-Q plot) can be constructed only for distributions of reversible cumulative distribution functions (CDF) such as the exponential distribution and the Weibull distribution. The CDF of generalised gamma distribution (GGD) is not only analytically irreversible, but also has no analytical form. However, at present, owing to advanced computer technology, this problem can be solved. The CDF of GGD can be inverted by using different computing environment, i.e. Microsoft Excel, Mathcad, R language.

In statistical literature there exist many tests to reveal the independence of two qualitative variables in two‑way contingency tables (CTs), in particular in 2×2 CTs. In this paper four independence tests were compared. These are: the chi‑square test, being the most popular type of power divergence statistics; the modular test and the d‑square test, which is a modification of the Pearson's test; the logarithmic minimum test which is a new proposal. Critical values for the tests listed above were determined with the Monte Carlo method. In order to compare the tests, the measure of untruthfulness of H0 was proposed and the power of the tests was calculated.

In the statistical literature there are many test measures to study the independence of features in the two-way contingency tables. For statistical analysis, the family of six so-called "chi-squared statistic" was selected — including Pearson's X² statistics — and the proposal of the author in the form of modular statistics. In order to free themselves from the limitations of the applicability of the "chi-squared statistic", critical values for all analyzed statistics were determined by simulation Monte Carlo methods. In order to compare the tests, the measure of untruthfulness of H₀ was proposed and calculated the power of the tests which is the ability of two-way contingency tables to reject null hypothesis which says that between features X and Y there is no relation.

Proposed by Pearson in 1900 ?2XY formula is still the most important measure to study the characteristics independence, especially since it has its extension for three variable and higher tables. The question is, what is the ability of two variable tables to detect relationship between features, what is their power. It is difficult to answer this question on the basis of the analysis. The best way seems to be generating two variable tables and determine power through simulation studies. For the 2x2 two variable table is it also possible to designate test power on the analytical way as well as comparison of obtained analytical results with empirical values. The work results will allow the reader to get an idea of the extent to which power of two variable tables depends on the sample size and the strength of the association between features. Aim of this study is to provide a ready computer implementation to test power of two variable tables stated as a set on the Internet. Presented theory and some examples will help readers to explore the test power using Pearson's ?2 statistics and model the course of the density function and cumulative distribution central and non-central chi-square distribution.

In the study of the independence of characteristics in the multi-feature tables X2 Pearson's statistics are the most popular. For multi-feature arrays, there are certain limitations as to the applicability of the X2 Pearson's statistics, but in an era of rapidly developing computer setting can be abolished with the critical value of computer simulation to generate the contents of multi-feature tables. The aim of the study is to provide the reader with a ready computer implementation, written in VBA editor (Visual Basic for Applications). Reading the presented theory for multi-feature tables and analysis of the examples allow the reader to carry out independent tests using X2 Pearson's statistics at any number of objects in each multi-feature table cells.

In the literature are popular statistical tool to measure the strength of the association between features in the 2-feature tables. However, the mechanisms to measure the strength of the association between the 3 feature in the tables are not too popular, and Polish literature practically does not exist. This paper describes an 3-feature array with its characteristics and provides 3-feature tables theory. The paper presents the computer implementation for measuring the strength of the relationship between the three modes with the Gray-Williams, Marcotorchino and Lombardo factors. The article also presents ready procedures and functions stored in VBA, enabling the reader to carry out independent surveys.

This paper discusses the theory of contingency tables with particular emphasis on three-dimension-tables. Their use in statistical surveys is stressed in the article. The three-dimension-table is presented as a chi-square test of independence and its implementation in VBA (Visual Basic for Applications) language. In this way, users are provided ready-made procedures and functions for carrying out surveys in Microsoft Excel. It has been examined also the ability of the three-dimension-table to detect the relationship between the measured traits, referred to as the power of the test.

This paper proposes a modification of the classic test of independence chi-square involving the use of modular statistics, as a measure of discrepancy between the observed and hypothetical distribution. Module statistics differs from Pearson chi-square statistics that deviation square of the theoretical and expected empirical probability is replaced by absolute value. For the proposed statistics is given the critical value |Χ|α α by different size samples of significance α∈{0,01; 0,05; 0,1}. The ability of two-part tables was examined to detect a relationship between characteristics, defined as the power of the test as well as the results were compared with the power of the test of independence, using statistics by Χ2 z (k-1)(w-1) degrees of freedom. The author examined whether the distribution of discrepancy measurements of two-part tables is consistent with the theoretical distribution of chi-square by Χ2 z (k-1)(w-1) degrees of freedom. Their usefulness in statistical practice was indicated too.

The article concerns two-way (2×2) contingency tables. When the H0— hypothesis of independence of features is correct, very often — because of the small sample — the distribution of the test statistics differ from the chi-square. Quantile of the chi-square is therefore not a correct critical value. With the current performance of computers, designation of critical value by statistical modeling of Monte Carlo method is not a problem, but a problem is H0 modeling. The H0 modeling is generating such arrays, which feature value assigned rows are independent of the characteristics of the assigned columns. Suitable for such modeling are tables — uniform of the same probability of belonging to cells and uneven having equal probability in all rows of a given column or in all columns of a given row. Analysis of the results of statistical modeling revealed that even when H0 is right, the distribution of the test statistics significantly depends on the uneven array. The article shows that in order to maximize the power of the test should be set critical value, taking into account measures of inequality array. The final result of the study is offered the reader a ready tool for independent verification of the H0 hypothesis.