Men vs. women. Blacks vs. whites. Rich vs. poor. Muslim vs. Christian. We hear a lot, in the social sciences and in the popular media, about how different various groups of people are in their preferences, traits, or behaviors. The finding of a “difference” based on empirical research is considered interesting and publishable! But it also, alas, often leads to much misunderstanding, and even invidious stereotyping.
This is because differences get a lot more attention than similarities. Because similarities are rarely reported on, we have a tendency to slide into thinking that differences are much larger than they actually are. It’s an easy slide from categorizing people under some labels—for example, drawing on people’s self-identification as “a woman,” “a man,” “white,” or “a person of color”—to thinking that traits and behaviors divide easily into the same categories.
Yet this is usually not the case! Humans are wildly diverse and individual, meaning that within groups, there is commonly a great deal of variation. Men vary from other men, rich people from other rich people, and so on. Meanwhile, comparing across groups, the fact that humans also share a common humanity means that many individuals in one group may feel or act in the exactly same way, in some situation, as many people in another group.
In my book Gender and Risk-Taking: Economics, Evidence, and Why the Answer Matters (2017) I show how supposedly scientific research on that particular topic has, though the neglect of similarities, tended to reinforce stereotypes. I also outline ways of correcting the research through use of several statistical techniques and greater care in the use of language. In this blog post, I would like to show you how just one of these techniques—one that I adapted from other contexts to use for the sorts of questions we are discussed here—can help shed light on the matter.
The Index of Similarity (IS) is an easily computable and understandable measure of the degree of overlap between two distributions. It is an especially useful tool when we are asking questions that require yes/no answers, or, say, rankings on a scale from 1 to 5. I’ll give a simple example to show it’s usefulness first, and then follow with a few technical notes.
Suppose that you ask a equal numbers of (self-identified) men and women whether or not they enjoy dancing. Suppose, hypothetically, that 70% of the women, but only 40% of the men, answer “Yes, I enjoy it.” Because 70% is not equal to 40%, you might want to conclude that “Men and women are different in their feelings about dancing” or “Women like dancing more than men.” But are these men and women also similar in their feelings about dancing? Yes!
IS measures the degree of overlap between two distributions—that is, the proportion of people in one group whose behavior matches up with that of people in the other group. IS takes on a value a value of 0% if the distributions don’t overlap at all, and a value of 100% if the distributions are totally identical. In between those extreme values, difference and similarity coexist.
The picture below illustrates this for the current example. If 70% of women like dancing, that means that the remaining 30% do not. These latter women can be easily paired with corresponding men. Likewise, the 40% of men who do like dancing can be easily paired with corresponding women. Overall, the overlap between the two distributions, which is IS, is 70%. The degree of difference is the remaining 30 percentage points.
We may, in fact, say that these two groups are more similar than different. Were we to picture these men and women attending a heterosexual singles social event, we might imagine 30% happily paired in conversations on the sidelines, 40% happily paired up out on the dance floor, and only the remaining 30% finding themselves unhappily paired with a partner who would rather be doing something else.
If a researcher simply reports that “Women like dancing more than men” without also reporting how similar the distributions are, misunderstandings easily arise. We might think they mean that the groups are categorically different—that is, IS = 0% and nobody at the social event would be happy. We might call this the “Mars versus Venus” view, after the book that claimed that men and women are so different as to inhabit different planets. But we can also say that a difference exists when IS takes any value short of exactly 100%. The claim of “difference” could still be made when IS = 99%, and nearly everybody at the social event finds an amenable activity partner. Instead of Mars versus Venus, the truth may be “North Dakota versus South Dakota”—yes, different, but mostly similar. The meanings of the statements about “more” or “different,” unqualified by any indication of the degree of similarity, are wildly ambiguous.
Because our brains find it very easy to categorize and stereotype, however, we tend to gravitate towards exaggerated “difference” beliefs when faced with this ambiguity. While the dancing example is trivial, the real world consequences of neglecting similarities are not. In my work on gender differences in preferences concerning economic and financial risk-taking, I found researchers claiming to find “fundamental differences” between the sexes—even though, when I calculated it, IS frequently exceeded 90%. In my broader reading, I have found people expressing unfounded beliefs about men’s and women’s suitability for various jobs, the willingness of rich versus poor people to obey laws, and the likelihood that a black versus a white person will have criminal tendencies, all based on exaggerated “difference” claims. Using IS gives similarity a chance to be recognized, too.
When responses come in a more complicated form, such as “on a scale of 1 to 5” we need more general equations. In the more general case, IS can be calculated as
where f i/F is the proportion of females within category i, and m i/M is the proportion of males in that same category. As explained above, it has an intuitive interpretation as the proportion of the females and males (in equal-sized groups) that are similar, in the sense that their characteristics or behaviors (on this particular front) exactly match up with someone in the opposite sex group. The second, equivalent formulation above is the discrete case of what has been called the “overlapping coefficient,” and corresponds to the intuitive explanation illustrated in the figure above: IS is equal to the sum of the lower bars in a histogram of the discrete distributions. IS can be easily computed from empirical data with a few lines of programming in software such as Excel or Stata.
What if the data are continuous, or the number of categories is quite large? Unfortunately, IS may be less helpful in such cases, since it will be sensitive to the ranges used to create groupings. See my book or Hyde (2005) for explanation of how to investigate similarities using Cohen’s d and related statistics. If one has large sample sizes, one might also investigate non-parametric estimation techniques for the overlapping coefficient that are said to have good asymptotic properties (Anderson. Linton et al 2012).
Anderson, Gordon, Oliver Linton, et al. (2012). “Nonparametric Estimation and Inference about the Overlap of Two Distributions.” Journal of Econometrics 171: 1-23.
Hyde, Janet Shibley (2005). “The Gender Similarities Hypothesis.” American Psychologist 60(6): 581-592.
Nelson, Julie A. (2017) Gender and Risk-Taking: Economics, Evidence, and Why the Answer Matters. NY: Routledge.