Sociology and Probability Theory

Abstract

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).

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 to 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 understand the influence of probability on statistics and the abilities and limits of statistics.

Hypothesis Testing. A hypothesis is an empirically verifiable declarative statement that the independent and dependent variables and their corresponding measures are related in a specific way as proposed by a theory. The independent variable is the variable that is manipulated by the researcher. For example, if a behavioral researcher wants to know if people are treated differently at work depending on how they dress, the independent variable in this hypothesis is the type of attire that people wear to work. The dependent variable—in this case, the way people are treated at work—is the variable whose value depends on the value of the independent variable.

For purposes of statistical testing, a hypothesis is stated in two ways. The null hypothesis (H₀) is the statement that there is no statistical difference between the status quo and the observations of the researcher after the manipulation of the independent variable. In other words, it states that the variable being studied (e.g., the way that people dress at work) has no bearing on the end result (e.g., the way that people are treated at work). This means that if the null hypothesis is true, the manipulation of the independent variable in an experiment does not change the results. The alternative hypothesis (H₁) states that there is a relationship between the two variables (e.g., people who dress in a businesslike manner get better treatment in the workplace). In general, inferential statistical tests are used to test the probability of the null hypothesis (H₀) being true.

When one accepts the null hypothesis, one is concluding that if the data in the underlying population are normally distributed, the results observed in a research study are more than likely due to chance. This is illustrated in Figure 1 as the unshaded portion of the distribution. Going back to the previous example, a researcher who accepts the null hypothesis is stating that it is likely that people who wear business suits and people who wear casual clothing are not treated any differently in the workplace. However, if the null hypothesis is rejected and the alternate hypothesis is accepted, the researcher is concluding that the results are unlikely to have occurred due to chance and are more likely attributable to some underlying factor. This conclusion assumes that there is a statistically significant difference between the way that the two groups are treated, due not to chance but rather to a real underlying difference in people's attitudes about how others dress in the workplace. Thus, since the results are statistically improbable, the differences observed in the study would lie within the shaded portions of the graph.

The results of statistical hypothesis testing are expressed in terms of probability, or the likelihood of an event occurring. Probability is the mathematical expression of the number of actual occurrences in relation to the number of possible occurrences of the event, expressed as a value between 0 and 1.0. A probability of 0 signifies that there is no chance that the event will occur, while 1.0 means that the event is certain to occur. When used within the paradigm of the scientific method, probability theory can be applied to real-world situations to estimate whether or not the observed results of the research are more likely to be due to an underlying cause or to mere chance.

Several steps are involved in this estimation. First, the researcher must define a population that has certain parameters. Continuing with the example of voting preferences used above, one might want to define the population as all voters within the United States, a two-part requirement: one part distinguishes between people who vote and people who do not vote, and the other part distinguishes between people who vote in the United States and people who vote in other countries that may have different political systems. Second, the researcher needs to draw a random sample from this defined population under the assumption that such a random draw will decrease the possibility of introducing bias into the sample. Third, the researcher needs to be able to reason probabilistically. In other words, he or she must be willing to run the risk of being wrong and must specify what an acceptable probability of being wrong is (e.g., 5 times out of 100). By specifying this probability, the researcher expresses the threshold of the probability of occurrence at which he or she would feel confident rejecting the null hypothesis. Finally, the researcher needs to set up a null hypothesis and an alternative hypothesis that cover the range of possibilities regarding the research question. The normal curve, as shown in Figure 1, represents all possible values for the results of the study. The shaded areas in the distribution represent the unlikely possibility that the results are not due to chance—typically a 1 or 5 percent probability—while the larger area represents the possibility that the observed results are due to chance.

Interpreting Statistical Results. As discussed above, statistics does not state with certainty whether or not the null hypothesis is correct; it only estimates the probability of it being correct. Therefore, because of the laws of probability, no matter whether one accepts or rejects the null hypothesis, there is always a possibility of error when interpreting statistics. When interpreting statistical results, two types of error are possible. Type I error, also referred to as an alpha (α) error, occurs when one incorrectly rejects the null hypothesis and accepts the alternate hypothesis. For example, a Type I error would have occurred if the researcher concluded that people who dress more formally in the workplace receive better treatment when, in fact, the way people dress has no effect on how they are treated. A Type II error, also referred to as a beta (β) error, occurs when one incorrectly accepts the null hypothesis. For example, a Type II error would have occurred if the researcher interpreted the statistical results to mean that people receive the same treatment in the workplace no matter how they dressed when, in fact, people who dress in a more businesslike manner tend to receive more advantages or rewards. As one decreases the likelihood of the alpha error, one concomitantly increases the likelihood of the beta error. However, this does not occur proportionately. The goal, therefore, is to determine the best balance between alpha and beta errors. For this reason, a p-value of .05 is usually taken as a minimum requirement for rejecting the null hypothesis. The conditions of Type I and Type II errors are shown in Table 1.

Conclusion

In order to better understand, interpret, and predict the phenomena around us, researchers cannot rely solely on their own experience or motivations; they must apply probability theory in the form of inferential statistics to the data they collect. 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. Without an understanding of probability theory and what statistics can and cannot accomplish, it can be tempting to look at the results, perform a few descriptive statistical techniques, and draw a conclusion about whether or not one's hypothesis is true. Such results tend to be specious, however. It is the application of the scientific method, including statistical analysis, that determines whether a body of knowledge is philosophy or science.

Terms & Concepts

Bias: The tendency for a given experimental design or implementation to unintentionally skew the results of the experiment due to a nonrandom selection of participants.

Data: In statistics, quantifiable observations or measurements that are used as the basis of scientific research.

Descriptive Statistics: A subset of mathematical statistics that describes and summarizes data.

Distribution: A set of numbers collected from data and their associated frequencies. A normal distribution is a continuous distribution that is symmetrical about its mean and asymptotic to the horizontal axis. The area under a normal distribution is 1. The normal distribution, also called the Gaussian distribution or the normal curve, describes many characteristics observable in the natural world.

Inferential Statistics: A subset of mathematical statistics used in the analysis and interpretation of data, as well as in decision making.

Null Hypothesis (H₀): The statement that the findings of an experiment will show no statistically significant difference between the control condition and the experimental condition.

Population: The entire group of subjects belonging to a certain category, such as all women between the ages of 18 and 27, all dry-cleaning businesses, or all college students.

Probability: A branch of mathematics that deals with estimating the likelihood of an event occurring. Probability (p) is expressed as a value (p-value) between 0 and 1.0, which is the mathematical expression of the number of actual occurrences compared to the number of possible occurrences of the event. A probability of 0 signifies that there is no chance that the event will occur, while 1.0 signifies that the event is certain to occur.

Sample: A subset of a population. A random sample is a sample that is chosen at random from the larger population with the assumption that it reflects the characteristics of the larger population.

Scientific Method: General procedures, guidelines, assumptions, and attitudes required for the organized and systematic collection, analysis, and interpretation of data that can then be verified and reproduced. The goal of the scientific method is to articulate or modify the laws and principles of a science. Steps in the scientific method include problem definition based on observation and review of the literature, formulation of a testable hypothesis, selection of a research design, data collection and analysis, extrapolation of conclusions, and development of ideas for further research in the area.

Statistics: A branch of mathematics that deals with the analysis and interpretation of data. Mathematical statistics provides the theoretical underpinnings for various applied statistical disciplines, including business statistics, in which data are analyzed to find answers to quantifiable questions. Applied statistics uses these techniques to solve real-world problems.

Variable: An object in a research study that can have more than one value. Independent variables are stimuli that are manipulated in order to determine their effect on the dependent variables, or response variables. Extraneous variables are variables that affect the dependent variables but are not related to the question under investigation.

Bibliography

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Suggested Reading

Egecioglu, Ö., & Giritligil, A. E. (2013). The impartial, anonymous, and neutral culture model: A probability model for sampling public preference structures. Journal of Mathematical Sociology, 37, 203–222. Retrieved November 6, 2013, from EBSCO Online Database SocINDEX with Full Text. http://search.ebscohost.com/login.aspx?direct=true&db=sih&AN=90274270&site=ehost-live

Ghilagaber, G. (2005). Incompatibility between hazard- and logistic-regression in modeling survival data with multiple causes of failure. Quality & Quantity, 39, 37–44. Retrieved April, 7 2008 from EBSCO Online Database SocINDEX with Full Text. http://search.ebscohost.com/login.aspx?direct=true&db=sih&AN=16525781&site=ehost-live

Sartori, R. (2006). The bell curve in psychological research and practice: Myth or reality? Quality & Quantity, 40, 407–418. Retrieved April, 7 2008 from EBSCO Online Database SocINDEX with Full Text. http://search.ebscohost.com/login.aspx?direct=true&db=sih&AN=20907730&site=ehost-live

van den Berg, G. J., Lindeboom, M., & Doton, P. J. (2006). Survey non-response and the duration of unemployment. Journal of the Royal Statistical Society, 169, 585–604. Retrieved April 7, 2008 from EBSCO Online Database SocINDEX with Full Text. http://search.ebscohost.com/login.aspx?direct=true&db=sih&AN=21097418&site=ehost-live

Yang, X.-S., & Jeliazkov, I. (2014). Bayesian inference in the social sciences. Hoboken, New Jersey: Wiley. Retrieved October 30, 2018, from EBSCO Online Database eBook Collection (EBSCOhost). http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=888972&site=ehost-live&scope=site