A ( brief ) Introduction to Inferential Statistics

The eld of statistics can be divided into two branches, descriptive statistics and inferential statistics. Descriptive statistics, as its name suggests, is concerned with describing a population, or a sample from that population, in terms of numerical measures that are computed from data. Such a numerical measure is called a statistic. If the data used in the computation comes from the whole population, then the statistic is called a population statistic. Likewise, if the data comes from a sample of the population, then the statistic is called a sample statistic. For example, the average height of all UCSC undergraduates is a population statistic, while the average height of 300, randomly selected undergraduates is a sample statistic. It is often di cult or impossible to collect accurate data from an entire population, making it di cult or impossible to compute the relevant (population) statistics. Thus, the practice is to collect data from a sample of the population, compute the relevant sample statistics and use these to draw inferences about the unknown population statistics. Inferential statistics is the branch of statistics that uses mathematical methods, of which probability theory is the main component, to draw inferences about population statistics based on the corresponding sample statistics. The value of a population statistic, though unknown, is constant, assuming that the population doesn't change. On the other hand, the value of a sample statistic typically varies from one sample to the next, and this value is uncertain until speci c sample data are collected. In other words, a sample statistic is a random variable, whose value depends on the particular sample that is chosen. The probability distribution of this random variable is called the sampling distribution. For example, suppose that there are currently 10000 undergraduates at UCSC, then their average height is some constant H . On the other hand, there are