Distribution Function of Statistics and Hypothesis Testing

Abstract

Every quantity that is estimated from the data, such as the mean or the variance of a Gaussian variable, is subject to statistical fluctuations of the measurements. For this reason, they are referred to as a statistics. If a different sample of measurements is collected, statistical fluctuations will certainly give rise to a different set of measurements, even if the experiments are performed under the same conditions. The use of different data samples to measure the same statistic results in the determination of the sampling distribution of the statistic, to describe what is the expected range of values for that quantity. In this chapter we derive the distribution of a few fundamental statistics that play a central role in data analysis, such as the χ ² statistic. The distribution of each statistic can be used for a variety of tests, including the acceptance or rejection of the fit to a model, as explained in later chapters.