## Formulas

## Formulas

# Formulas

Although most social science research concerns characteristics of people, groups, organizations, and situations for which there is no inherent metric or measure, virtually all such research involves quantifying these characteristics. The data that result from this quantification are manipulated, analyzed, and interpreted using mathematical formulas. Formulas also are used by social scientists to describe or model behavior, particularly interpersonal behavior, using mathematical operations and principles. In some social science research literatures, such formulas are the principle means by which behavior is described and predicted.

William James, in his *Principles of Psychology* (1890), offered one of the earliest examples of a description of an aspect of human social experience in mathematical terms. James suggested that self-esteem is a function of the ratio of people’s successes to their “pretensions”:

In other words, a person’s self-esteem depends on the degree to which they are meeting their own expectations. Inspired by James’s work, psychologists a century later posited more complex formulas in which self-esteem is a measure of the sum of people’s performances in various domains, weighted by the importance they ascribe to those domains.

An area of social science in which mathematical formulas have been particularly influential is interpersonal relations. Virtually all of these formulas derive in part from social exchange models of interpersonal behavior. The seminal social exchange account of interpersonal relations was proffered by J. Stacy Adams in 1963. Adams’s specific concern was the degree to which individuals feel as if they are treated in a just and fair manner by their employers. According to Adams, perceptions of just and fair treatment stem from perceptions of equity, which can be modeled using the following formula:

Inputs are what the employee puts into the job (e.g., effort, loyalty), and outputs are what the employee gets in return (e.g., salary, job security). As this ratio departs from 1.0, particularly in the direction of inputs exceeding outputs, perceptions of fairness, as well as motivation, decline.

An important addition to this simple model is an accounting for the role of comparisons in such judgments. For instance, the equity ratio might be expanded to:

In this case, “other” could denote another person (e.g., a coworker) or another opportunity (e.g., a position with another employer). The addition of comparisons yields a model that can account for the fact that individuals sometimes remain in relationships despite inputs that exceed outputs, or that they leave relationships that are providing outputs in excess of inputs.

Other models inspired by the social exchange perspective use formulas as a basis for defining and predicting relationship outcomes. For instance, Caryl Rusbult’s investment model defines relationship satisfaction using the following formula:

Satisfaction = (*rewards* – *costs* ) – *comparison level*

This formula specifies that people experience satisfaction in relationships when the difference between rewards and costs in the relationship exceeds expectations. Satisfaction, defined in this way, is a term in the formula for defining commitment to the relationship:

Commitment = *satisfaction* – *alternatives* + *investments*

“Alternatives” corresponds to the perceived degree of satisfaction the individual could expect to experience in other relationships, and “investments” correspond to the accrued costs of staying in the relationship (e.g., opportunities not pursued in order to preserve the relationship). These formulas define relationship outcomes and suggest means by which they can be predicted and influenced, illustrating the strategic use of formulas in social science research.

Other theoretical models in the social sciences specify formulas for which specific values of terms are predicted. An instance of such a use of mathematical formulas is Bibb Latané’s social impact theory. According to this theory, the influence of other people on an individual can be specified in terms of three factors: (1) *strength*, or how important the individuals are; (2) *immediacy*, or how close in space and time the individuals are; (3) *number*, or how many people there are. Social influence, or impact, is a product of these factors, so that:

Impact = *strength ^{i}* ×

*immediacy*×

^{i}*number*

^{i}For example, the model predicts that performance anxiety will increase as the number of individuals watching, their importance to the performer, and their proximity to the performer increases. An important feature of this model is the assumption that the influence of these factors is not linear (i.e., the exponents fall between 0 and 1). Take for instance, the number factor. The model specifies that, holding strength and immediacy constant, the addition of one more person in the situation has less influence if a large number of people are already present. Likewise, the addition of one more person in the situation has more influence if a small number of people are already present. Thus, associated with each factor is an exponent that describes in mathematical terms the relation between that factor and its impact.

Even in social science research for which concepts are not framed in mathematical terms, mathematical formulas are important, because the data generated by social science research is virtually always subjected to statistical analysis. The most prevalent use of formulas in this context is for the construction of test statistics. Test statistics, such as *t, F,* and *x* ^{2}, are used in hypothesis testing as a means of evaluating the likelihood that an observed pattern of results is attributable to chance. This likelihood is reflected in the *p* -values that accompany observed values of test statistics, with values lower than .05 indicating, by convention, statistical significance.

A related use of formulas is for the computation of effect sizes, which are means of indexing the practical, as opposed to statistical, significance of a research finding. A frequently used effect size, *d,* is attributable to the American psychological researcher Jacob Cohen. The formula for computing *d* is:

Here, *M* _{1} and *M* _{2} are means on some outcome for two groups and σ is a standard deviation, either for one of the groups or for a “pooled” standard deviation. The resultant value is the difference between the two groups in standard deviation units, which is interpreted with reference to criteria for small, medium, and large effects, as described by Cohen. Other effect sizes can be generated using formulas specific to the statistical model used to analyze the data.

Mathematical reasoning is a routine activity in quantitative social science research. At the most fundamental level, mathematical formulas are, in some instances, used to define and predict variables. In all cases, mathematical formulas are used to construct test statistics required for hypothesis testing. Increasingly, these test statistics are accompanied by effect sizes, which make use of output from statistical analyses to construct indexes of practical significance. For these reasons, social scientists routinely use mathematical reasoning in their work.

**SEE ALSO** *Methods, Quantitative; Models and Modeling; Quantification; Social Science; Statistics; Statistics in the Social Sciences*

## BIBLIOGRAPHY

Adams, J. Stacey. 1963. Toward an Understanding of Inequity. *Journal of Abnormal and Social Psychology* 67: 422–436

Cohen, Jacob. 1988. *Statistical Power Analysis for the Behavioral Sciences*. 2nd ed. Mahwah, NJ: Erlbaum.

Kelley, Harold H., and John Thibaut. 1978. *Interpersonal Relations: A Theory of Interdependence*. New York: Wiley-Interscience.

Rusbult, Caryl E. 1980. Commitment and Satisfaction in Romantic Associations: A Test of the Investment Model. *Journal of Experimental Social Psychology* 16: 172–186.

*Rick H. Hoyle*