Introduction

The Test of Normality is one of the most important statistical hypothesis tests, as it determines the probability that a random variable can be adequately described by the Normal Distribution. The Normal Distribution holds a central place in Statistics, since many analytical methods are based on it as a core assumption. For this reason, the test of normality usually precedes statistical data analysis and determines whether parametric or non-parametric methods should be applied for hypothesis testing and further analysis.

Importance of the Test of Normality

The importance of the normality test extends both to theoretical and applied statistics. At the level of Descriptive Statistics, the test measures the degree of fit of a normal model to the data and provides a clear picture of their shape and characteristics. In Inferential Statistics, the test examines whether the data can be considered normally distributed and, therefore, whether the conditions for applying parametric tests are met. By contrast, in Bayesian Statistics, there is no formal test of normality; instead, probabilities are calculated for the data to originate from a normal distribution compared with the probability of originating from alternative distributions, using the Bayes factor.

Methods of Testing Normality

One of the most well-known methods is the Kolmogorov-Smirnov test, which is non-parametric and compares the empirical cumulative distribution function of a sample with the theoretical distribution of the null hypothesis. This theoretical distribution may be normal, but also uniform, Poisson, or exponential. The Shapiro-Wilk test is another important normality test, particularly effective for small and medium-sized samples. It assesses whether observations originate from a normal distribution by applying specific coefficients, and due to its statistical power it is considered one of the most reliable tests.

Beyond numerical tests, graphical methods also play a key role. The P-P Plot (Probability-Probability Plot) compares the cumulative probabilities of observed data with the theoretical probabilities of a distribution. If the points converge along the straight diagonal line, then normality is likely. The Q-Q Plot (Quantile-Quantile Plot) compares the quantiles of the data with the quantiles of a theoretical distribution, allowing for the examination of location, scale, and skewness. This visual approach is particularly useful for identifying deviations and for understanding whether the data follow a normal shape or present irregularities.

Interpretation of the Tests

The interpretation of normality tests varies depending on the context of the analysis. In Descriptive Statistics, the aim is to assess how well a normal model fits the data, without necessarily evaluating the underlying randomness. In Inferential Statistics, the focus is on determining whether the null hypothesis of normality should be rejected or not. In Bayesian Statistics, probabilities are compared to estimate the suitability of different distributions in describing the data.

Relationship with Limit Theorems

The Central Limit Theorem (CLT) is one of the main reasons why normality holds such a central position. According to the theorem, regardless of the original distribution of the data, the distribution of sample means approaches the Normal Distribution as the sample size increases. However, there are cases where the normality of the sampling distribution is not sufficient, and the normality of the actual data is required, particularly when the samples are small and when statistical methods sensitive to deviations from the Normal Distribution are applied.

Conclusions

In conclusion, the Test of Normality is essential in statistical analysis, as it guides the choice of the appropriate model and directly affects the validity of conclusions. Through numerical tests such as Kolmogorov-Smirnov and Shapiro-Wilk, as well as graphical methods such as P-P and Q-Q plots, researchers can determine whether data fit the Normal Distribution adequately. Their application, combined with the theoretical basis of the limit theorems, offers a powerful tool for understanding and properly processing statistical data, thus allowing for the selection of the most suitable analytical methods.