At the beginning of a study, the descriptive statistics can provide a summary of essential information about the quantitative data that is being examined in more detail in the body of the study. Descriptive statistics simply summarize facts about the data that has been collected, and do not support inferences or hypotheses. The overall population size and the sizes of subgroups within the population and descriptive numerical information about the subgroups is provided. The number of males and females in the population. The numbers of individuals who fall into age groups, ethnic or racial groups, and other category data is summarized when those factors are relevant to the study.
When there is a hypothesis that a difference between two groups is the result of a variable and not from random factors, a test to determine the strength of the difference is done, and the results are included in the descriptive statistics for the study. A complicated transaction called the P Value, which tests whether the null hypothesis, or probability that the data does not support the hypothesis is a critical descriptive statistic, since it tells whether the hypothesis is weak.
There are measures to identify the amount of central tendency, association and dispersion of the observations. There are various graphic representations of the data, such as histograms, scatter plots, and diagrams.
The classic Standard Deviation From The Mean, or Standard Score describes the distance of a particular observation from the mean for the entire population. This is a process called "normalization", which generally tells how the population and individuals observations are distributed in relation to the population mean. There are a variety of tests or scores to determine the ways in which the population clusters around or deviates from a variety of standardized positions that relate to the mean. This process can be used to determine how subsets of the population differ in relation to the whole or to each other. When the entire population is not known, normalization attempts to determine the ability of the sample to represent the whole. When there is a causative relationship between two variables, the strength of the relationship can be tested.
Mathematical statistics is where the fun begins, and where inferences and deductions are tested with quantitative data. Inferential statistics involves hypothesis testing, testing of probability, and estimations of points and intervals.
Mathematical statistics can be in the form of number theory, where each observation is considered as a dot on a graph. When analyzed in this way, various formulas can be applied to identify differing patterns, underlying patterns, and even as layered patterns in three dimensional form.
In probability mathematical statistics, the two main schools, Bayesian and Frequency, must be understood. The Frequency school adheres to a tight reliance on numbers and their frequency as the basis for making conclusions about probability. The Bayesian school has a looser standard of "a degree of rational belief." 1
Both descriptive and mathematical statistics are areas that are being increasingly misused, abused, and misunderstood. It is essential to plow through and to get a good understanding of basic and even advanced statistical theory, philosophy, and analysis in order to understand what the information truly means, and how comprehensive, accurate, and reliable the raw data as well as the application of various statistical modeling is.
Martin Serenstein, PhD "Statistics", 1996, Barrons Educational Series, Inc