## Sociology & Probability Theory

Probability theory is a branch of mathematics that deals with the estimation of the likelihood of an event occurring. An integral part of the scientific method and the principles of probability theory are applied through inferential statistical techniques in the analysis of data to determine the likelihood that a hypothesized relationship between variables exists. Applied probability theory, however, does not "prove" that a hypothesis is correct: it only expresses the confidence with which one can state that variables are related to each other in a way stated by a theory. Even when the results of a research study are statistically significant, the researcher still accepts the possibility of error by either rejecting the null hypothesis when it is, in fact, true (Type I error), or accepting the null hypothesis when it is, in fact, false (Type II error).

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ACADEMIC TOPIC OVERVIEWS

Research Methods > Sociology & Probability Theory

### Overview

It is typically safe to assume that the majority of students studying sociology or other behavioral and social sciences are not doing so primarily because of their interest in pure mathematics. People interested in math and science are usually more immediately drawn to the "hard" sciences, where things can be weighed, measured, and counted and where the difference between a score of 12.00 and a score of 12.01 has a tangible meaning. People drawn to the social and behavioral sciences tend to be more interested in the wide variety and great unpredictability of human behavior. However, mathematical and scientific tools are essential to sociologists and other behavioral scientists as they seek to understand, interpret, and predict the behaviors that they see around them.

Unfortunately, too often the social sciences are considered by outsiders to be nothing more than the mere articulation of "common sense." Yet common sense is often not at all common, and examining one's own motivations and behaviors yields only limited insight into the motivations and behaviors of others. I may know why I act the way I do in certain situations, but it is not logically valid to generalize my knowledge about my opinions or behavior to apply to other people.

For example, I may have strong feelings about a certain political candidate and cast my vote accordingly. It might seem, therefore, that everyone confronted with the same evidence should vote the same way I do. However, neither horse races nor elections are that easy to predict. I cannot simply extrapolate my opinions and voting behavior to predict how others will vote. In order to do so, I must understand my deeper motivations, as well as the factors that impact others' decisions. For example, I may favor a certain candidate because of a speech he or she made, an article that I read in the newspaper, or the candidate's record in previous elected positions. However, it is unwise to base my vote on one speech, article, or even record. Speeches tend to be stylized and without deep content, newspaper columnists often have their own agendas and fail to be objective, and public records do not necessarily reflect how a candidate will perform if elected into a different office, or if he or she has changed his or her mind about a political issue. Someone who disagrees with my choice of political candidate may possess some of these missing pieces of information and, therefore, reach a different conclusion than mine. Alternatively, the person disagreeing with me may not have all the information that I possess. Social scientists try to unravel these and other complex issues by studying large populations of people.

When one talks about human behavior, one must consider other factors as well. Human beings cannot necessarily judge evidence objectively. One person may always vote for candidates in a certain political party because his or her parents did so. Another person may vote for a political party because he or she dislikes the general platform of the opposing party. Someone else may believe that a candidate belonging one social group could not possibly understand or fairly represent the needs of another social group and vote accordingly. Still others might vote for a candidate because of the way the candidate dresses or because he or she appears to be a "nice" person.

Because the human decision-making process is so complex, it is virtually impossible to generalize from the opinions and behaviors of one person, or even a small group of people, to make conclusions about how all human beings think or behave. For this reason, it is important to use the scientific method and probability theory to better understand why people act the way they do. Without an understanding of probability theory, one can easily fall into the trap of thinking that a single set of statistics "proves" a theory, or that the results of one study will be replicated in all future studies on the same topic. Because it is impossible to extrapolate from the behaviors and motivations of one individual to the behaviors and motivations of individuals in general, it is necessary to use scientific and mathematical tools to better interpret the world around us. Probability theory, and the inferential statistical tools that are based on it, helps scientists and researchers determine whether the results that they observe are due to chance or to some other underlying cause.

### Applications

Virtually every semester, at least one of my students proudly announces that the statistical analysis he or she has performed on his or her research data "proves" that his or her hypothesis is right. As satisfying as this conclusion might be, in truth, statistics do not prove anything, nor is the scientific method a quest for proof. Statistics merely express confidence and describe probabilities concerning whether or not the null hypothesis is more likely to be true than the alternate hypothesis. This fact is frequently demonstrated in scientific literature when one group of scientists attempts to replicate the research of another group and finds that their research results lead to conclusions that are different from the original group's. A lack of understanding of the way that probability works can lead to poor experimental design and spurious results. The results of a statistical data analysis do not prove whether or not one's hypothesis is true, only whether or not there is a probability of the hypothesis being true at a given confidence level. So, for example, if a t-test or analysis of variance yields a value that is significant at the p = .05 level, this does not mean that the hypothesis is true; it means that the analyst runs the risk of being wrong 5 times out of 100.

### Inferential Statistics

Without an understanding of probability theory and what statistics can and cannot accomplish, it may be tempting to look at the results of a research study or experiment, apply a few descriptive statistical techniques, and draw a conclusion about whether or not one's hypothesis is true. However, the objects of scientific study rarely yield black-and-white results. Even in the physical sciences, results can vary depending on the conditions under which a study was done. Therefore, inferential statistics are used to test hypotheses to determine if the results of a study have statistical significance, meaning that they occur at a rate that is unlikely to be due to chance, and to evaluate the probability of the null hypothesis (H0) being true. Inferential statistics allow the researcher to make inferences about the qualities or characteristics of the population that are based on observations of a sample. However, to understand what the results of statistical tests mean, one needs to...

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