Are Women Really More Risk-Averse Than Men

Are Women Really More Risk-Averse than Men? Abstract: While a substantial literature in economics and finance has concluded that women are more risk averse than men, this conclusion merits reconsideration. Drawing on literatures in statistics and cognitive science, this essay discusses the important difference between drawing conclusions based on statistical inference, which concerns aggregates such as mean scores, and generalization, which posits characteristics of individuals classified into kinds. To supplement findings of statistical significance, quantitative measures of substantive difference (Cohen's d) and overlap (the Index of Similarity) are computed from the data on men, women, and risk used in 28 published articles. The results are considerably more mixed and overlapping than might be expected. Paying attention to empirical evidence that challenges subjective cultural beliefs about sex and risk has implications for labor economics, finance, and the economics of climate change. 1 Are Women Really More Risk-Averse than Men? Julie A. Nelson 1 The Issue "We find that women are indeed more risk averse than men" conclude economists Rachel Croson and Uri Gneezy (2009, 448) in their Journal of Economic Literature review article "Gender Differences in Preferences." They base their conclusion on a number of empirical studies that found statistically significant differences between men's and women's behavior, on average, in lottery experiments or investment strategies—and in fact very similar statements are endemic in the literature. 2 But consider these two statements: A. "In our sample, we found a statistically significant difference in mean risk aversion between men and women, with women on average being more risk averse." B. "Women are more risk averse than men." While the two statements are often taken as meaning the same thing, there is in fact a wide gulf of meaning between statement A, which is a narrow statement that can be factually correct within the confines of a particular study, and statement B which is a broad statement that implies a stable characteristic of people according to their sex. Distinguishing between the two statements is important, since perceptions of sex differences in risk-taking have become part of both public and academic discussions about financial market stability (e.g., Kristof 2009); labor market, business, and investment success (e.g., Eckel and Grossman 2008, 14; Booth and Nolen 2012, F56); and environmental policy (e.g., Kahan, Braman et al. 2007). Broad statements about sex differences are also commonly made regarding other behaviors, such as competitiveness or management style. On a methodological plane, understanding the distinction between Statement A and Statement B is, in fact, important for understanding what we can really know based on statistical inference, in any application. The current essays explores this gulf primarily from a statistical point of view, but also touches on the linguistic, cognitive, epistemological, ideological, and policy-making causes and implications of sliding between two statements such as A and B. It is assumed 1 Funding for this project was provided by the Institute for New Economic Thinking. Matthew P. H. Taylor supplied expert research assistance. Many authors (as listed in footnotes) graciously shared with us the source data or specific statistics from their studies, for further analysis. 2 Statements of the form "women are more risk averse than men" occur in, for example, Arano, Parker et al. (2010), Bernasek and Shwiff (2001), Booth and Nolen (2012), Borghans, Golsteyn et al. (2009)—and nearly every other article on risk in the reference list for this essay. 2 that one goal of an economic science that aspires to objectivity is to make sure that the inferences drawn from statistical studies reflect only the statistical results, and not the influence of subjective factors such as individually-held preconceptions or culturespecific beliefs. The essay shows how the accuracy and objectivity of economic research could be improved, both by expanding our toolbox of quantitative methods and by improving the precision with which we interpret and communicate the results of our econometric research projects. What does "Women are more risk averse than men" mean? What might the statement that "women are more risk averse than men" mean? While it generally appears in the literature as though it is summarizing empirical results, recent research in cognitive science indicates that the relationship between statistical inference and what is communicated by this phrase is not at all simple or direct. The Statement Is Often Understood as Universal or Generic Taken as a bald statement, the statement might be taken as indicating something that is universally true for every individual member of the classes "women" and "men." In this case, it would have to be true that every individual woman is more risk averse than every individual man. This exceedingly strong implication is not likely intended by those who write such statements, since just one example of a cautious man and a bold woman disproves it. Cognitive research, however (reviewed below), suggests that many endusers of research may understand the statement in this way. A second possibility is that the statement is intended as what psychologists, linguists, and philosophers call a generic noun phrase (Gelman 2005, 3). A generic noun phrase is a generalization that "refers to a category rather than a set of individuals" and "express[es] essential qualities" implying that a category "is coherent and permits categorywide inferences" (Gelman 2005, 3). In a generic of the form "Fs are G," that is, "one is saying of a kind of thing, specified in the statement, that its members are, or are disposed to be G (or to [do] G) by virtue of being of that kind. The speaker conveys that being G is somehow rooted in what it is to be an F: G-ing is what Fs do (or are disposed to do) by virtue of being F" (Haslanger 2011, 13). Examples include "tigers have stripes." In this case, a few counterexamples—e.g., albino tigers—do not nullify the statement, since having stripes is considered to be part of the intrinsic, essential nature of tigerhood, even if stripes are not manifested in a particular case (Leslie forthcoming). "We essentialize a kind if we form the (tacit) belief that there is some hidden, nonobvious, and persistent property or underlying nature shared by members of that kind, which causally grounds their common properties and dispositions," writes Leslie (forthcoming). Generics seem to "articulate core conceptual beliefs" (Khemlani, Leslie et al. 2012, 1). In the current example, the statement would imply that greater risk-aversion is an essential characteristic of womanliness—or, by parallel reasoning, that greater risk3 seeking is an essential characteristic of manliness. 3 Such an essentializing view is evident in the risk and gender literature in article titles such as "Will Women Be Women?" (Beckmann and Menkhoff 2008) and "Girls will be Girls" (Lindquist and SäveSöderbergh 2011). The apparent presumption in such titles is that were a group of women or girls found to not be relatively more risk averse, they would somehow be abnormal relative to their own female natures. Many articles treat risk-aversion as a sex-linked "trait" (e.g., Powell and Ansic 1997; Borghans, Golsteyn et al. 2009)—presumably stable across time and cultural contexts. In addition, many studies hypothesize evolutionary explanations for female risk aversion or male risk-seeking (e.g., Olsen and Cox 2001; Hartog, Ferrer-i-Carbonell et al. 2002; Cross, Copping et al. 2011), or hypothesize links to sex-related hormones or other genetic factors commonly thought to define an essence of femaleness or maleness (see examples cited in Meier-Pesti and Penz 2008; Croson and Gneezy 2009). The belief that there is an "innate, genetic, or biological basis" for the statement is also a characteristic of many generics (Gelman 2005, 1). Generics Convey More (or Less) Than Statistical Prevalence Rather than being intended as universal or generic, a third possibility is that the general and qualitative statement "women are more risk averse than men" is simply meant as a convenient shorthand for the more nuanced and accurate statement that "women exhibit, on average, more risk-averse behavior than men in this particular study." One might suppose that the generic statement "Fs are G" might be functionally equivalent to quantitative statements such as "Fs are more G, on average," "most Fs are G," or "a majority of Fs are G." This turns out not to be the case, however, since the relationship between statistical prevalence and core beliefs expressed as generics is far from simple or direct. The more nuanced quantitative statement (e.g. "Fs are more G, on average") has been called an "aggregate-type proposition" in the statistical literature (Bakan 1955; Bakan 1966, 433). Unlike a generic, it implicitly acknowledges that not all women or all men are the same. Those who understand statistics will grant (at least upon reflection) that this means that the distributions of a characteristic among women and among men could overlap. The statement of an aggregate does not (logically) imply any story about causations or essences. An aggregate sort of statement can be empirically supported, within the context of a particular study. It expresses particular statistical results, not broad core beliefs. 4 The complicated relation between generic statements and quantitative statements has been examined in the psychological and philosophical literatures. While some generic statements seem to be accepted as true based on statistical prevalence (e.g., "cars have 3 The philosophical and linguistic literatures actually discuss various kinds of generics. For example, the generic statement "Mosquitos carry the West Nile virus" is generally considered true even though only a tiny minority of mosquitos carry the virus, due to the characteristic of carrying a disease being striking and dangerous (Leslie 2008). The essentialist version of a generic, however, is the one most relevant for the case examined here. 4 In the articles reviewed below, some examples of such more nuanced (and accurate) presentations of results can be found, even though generic statements predominate. 4 radios" discussed in Khemlani, Leslie et al. (2012)), and some research has suggested Bayesian models for the formation of generics (Khemlani, Leslie et al. 2012, 10), statistical prevalence cannot, in fact, explain many cases. Prevalence in a majority of a kind seems to be neither necessary nor sufficient for a generic to be considered true. For example, the generic statement "Ducks lay eggs" is generally accepted as true. In fact, only a minority of ducks (i.e., those that are female, mature, and non-sterile) lay eggs. We seem to reason that ducks are birds, and birds, as a "kind" or category, have the "characteristic" of reproducing by way of eggs (Khemlani, Leslie et al. 2012). On the other hand, "Canadians are right-handed" is rejected, even though a majority of Canadians are right-handed (Khemlani, Leslie et al. 2012). Rather than theorizing that generics are based in statistical regularities, then, a more promising line of research investigates the role of inductive reasoning and generalization in providing efficient (if far from fail-proof) modes of mental processing, as part of the process of human cognitive development (Gelman 2005; Leslie 2008). Empirical studies suggest that from a very early age, children create simple mental categories and classifications according to presumed essences (Leslie 2008). This propensity is so strong and basic that even among adults a statement phrased as a generic and accepted as true predisposes people to believe that individual members of a class will have the stated property (Khemlani, Leslie et al. 2009, 447)—that is, to interpret generic statements in ways that essentialize or even universalize the association. For example, given the statement "Quacky is a duck," people tend to agree with the statement "Quacky lays eggs" (Khemlani, Leslie et al. 2012, 10). One can also note that—tempting as it may be to think so—the truth of generics cannot be proven through statistical analysis. That is, evidence of an association between F (a kind) and G (a characteristic) in a statistical study does not provide an adequate basis for asserting that G is of the essence of F-ness. "Essences" and "kinds" are simply not the sorts of things that can be empirically observed in the outside world. Rather, the tendency to class things in terms of kinds seem to be part of the structure of our inside worlds— that is, of evolved, developmental human cognition. Correlations between biological genetic, hormonal, or evolutionary phenomena and differences in behavior by sex, for example, can be suggestive of "essential" differences, but are not conclusive evidence of essential differences. "Essential differences" may, for example, be only one among a number of possible explanations: Perhaps a third, confounding variable underlies the observed relationship between F and G. In regard to perceived differences by sex, for example, pressure to conform to prescribed social roles or relative positions in hierarchies of power have frequently been suggested as confounding variables (Acker 1990; Kimmel 2000; Barnett and Rivers 2004; Hyde 2005; Eliot 2009; Fine 2010). Another possibility is that further evidence may reveal that the behavior is not as closely tied to "kinds" as was earlier believed: Perhaps a broader sample of F will reveal evidence of not-G (e.g., the "black swan" observed after inductive reasoning led to the belief that swans are white, as discussed in Taleb (2010)). In the present context, for example, cross-cultural examination of sex differences in behavior may call beliefs about essences into question (Gneezy, Leonard et al. 2009; Blau, Ferber et al. 2010, 357; Else5 Quest, Hyde et al. 2010; Henrich, Heine et al. 2010). The fact that one may find explanations from biological or evolutionary "essences" plausible or even compelling does not mean that these explanations are true: Cognitive scientists use the term "confirmation bias" to refer to the human tendency to more readily absorb information that conforms to one's pre-existing beliefs. It has been suggested (Bakan 1966) that perhaps some confusion among researchers between inductive reasoning and Fisherian statistical inference may be behind considerable misinterpretation of statistical results. To reason inductively means to go from specific observations to hypothesizing general—i.e., universal or generic— propositions that invite conclusions about individuals or kinds. Fisherian inference, on the other hand, means going from sample results concerning a aggregate, such as a sample difference in means, to inferences about the corresponding population aggregate. Fisherian significance testing about a difference in means, for example, only (at best) justifies the inference that a difference in means in a sample corresponds to a difference in means in a population. It does not justify generalizing an (sample) aggregate statement to a generic or universal statement. Such issues have also been pointed out in the broader economics literature. Deaton's (Deaton 2009, 27-30) discussion of randomized control experiments in development economics makes a similar point: When all we get from a statistical study (no matter how well-designed) is information about differences in means, this does not justify inferences about other aspects of the distributions; does not supply the causal stories relating the variables of interest; and does not let us make predictions about individual events. Only when we read into the statistical results the existence of a generic relationship—e.g., "dams harm development" or "women are more risk averse"—are we tempted to, for example, make predictions about the next woman or the next dam. When Is a Difference of Substantive Importance? The implicit message behind a communicated finding of "difference"—especially if it is marked as the primary finding of a published article—is that the difference found is of a such a large substantive size that the result is worth reporting. As has been discussed to some extent in the economics literature, "substantive significance" and "statistical significance" are two different things (Ziliak and McCloskey 2004; Miller and Rodgers 2008). As is well known, differences can be statistically significant (i.e., generalizable to a larger population, under classical hypothesis testing) even if of small absolute size, especially in large samples. In the gender-and-risk literature, as in other literatures, however, judgments of "significant difference" are generally based on statistical significance alone. Discussions of the absolute size of the difference, much less its possible implications for society or policy, are rare. 5 This is in notable contrast to the psychological literature on behavior, where the expression of findings in terms of substantive difference is widespread, standardized, considered best practice (Wilkinson and Task Force on Statistical Inference 1999)—and where the implications for the study 5 In the literature reviewed here, Eckel and Grossmann (2008, 15) and Dhomen, Falk, et al (2011, 530, 540) are notable for providing any extended discussion of substantive economic significance. 6 of sex differences has been the topic of intense professional discussion (Eagly 1995; Hyde and Plant 1995; Archer 1996; Martell, Lane et al. 1996, and other articles in the March 1995 and February 1996 issues of the American Psychologist.). The default assumption that the difference must have been worth reporting, along with the abovementioned human tendency to essentialize or universalize based on generic statements, can be expected to contribute to the formation of (perhaps unfounded) beliefs that differences are substantively important, categorical, and possibly even nonoverlapping. Anecdotal evidence from this author's undergraduate classes suggests that statements such as "women are more risk averse than men" tend to often be understood as sharply disjunctive, along the lines of "women are risk averse and men are not." This seems to be especially true the closer the "essential" characteristic is associated with a biological factor: Students in these classes have been (at first response) unanimous and adamant in believing that the men and women have distributions of "the number of children given birth to" that are strictly non-overlapping. Presumably, if the abovementioned cognitive studies are correct, this is because the cognitively-available generic statement "Women bear children" tends to be more readily cognitively accessed than the empirical observation that some women, like men, never bear a child. Some of this tendency to neglect overlap can also be discerned in the risk literature. For example, as will be discussed below, two articles looking at financial risk concluded that females as a group have such a different attitude towards risk from men as a group, that they should always be matched with female financial advisors. The tendency to attribute dramatically distinct "essences" to men and women—so disjunctive as to make the two sexes as different as two different species—has been nicknamed the "Mars-versus-Venus" phenomenon, after the best-selling popular book Men are from Mars, Women are from Venus (Gray 1993). Examples of such assertions in the popular literature are now legion (see examples listed in Fine 2010). Substantive size, however, is eminently measurable and can be made a direct object of discussion for statistical researchers, and potentially (though perhaps with more effort) for the larger public. "Difference" and the Culture of Publication A further complication in interpreting this literature comes from the first phrase in the more precise formulation, "In our sample, we found a statistically significant difference in mean risk aversion between men and women…" Hypotheses about sex and risk aversion have clearly now been tested many times in the published literature—and, importantly, also in research findings that have remained unpublished. This raises the issue of what is known in meta-analytical statistics as the "file drawer effect." When a particular hypothesis is tested many times, statistically significant results will sometimes appear by chance even if a true difference is nonexistent or very weak (e.g. in the pure classical hypothesis testing case, in 10 studies out of 100 when α=.10). It is also quite clear that a statistically significant result is considered more 7 publishable than the finding of a lack of statistical significance. This biases the literature as a whole towards the reporting of difference, and against the reporting of similarity (as briefly mentioned in Croson and Gneezy 2009, 468). While the present study did not attempt to dig into file drawers, the analysis below suggests some publication bias towards the reporting of statistically significant differences only, even within the published works reviewed. Caution is Warranted In summary, the idea that a generic statement can serve as an innocuous shorthand for the accurate presentation of a study's results must be questioned, in the light of findings from cognitive science. Humans have a strong disposition to interpret generics as universals. Whether or not an economist who states "women are more risk averse than men" believes that there is some "essence" being communicated, the bald statement does much more than simply save the space that would have been used by inserting the qualifier, "on average": It tends to create an expectation among readers that that an individual woman can be predicted to be relatively risk-averse, and an individual man can be predicted to be relatively risk-loving. Other terms for this sort of false generalization are, of course, "stereotyping" and "prejudice." Given the human tendency to infer essences or universals from generic statements, a responsible researcher who wants to accurately present the results of his or her study, then, would be wise to confine himself or herself to precisely stated aggregate propositions, and discuss substantive as well as statistical significance. At a larger disciplinary level, these tendencies also suggest that reviewers and editors as well as authors should not overlook findings of similarity, if the creation of biases and distortions in disseminated knowledge is to be avoided. To be clear, the current essay is not arguing that “there is no difference between men and women," nor—by any means—does it take any stand concerning the existence or non-existence of metaphysical "essences." The concern here is with making the communication of empirical results conform to what can legitimately be inferred from the data. For accomplishing this, some expansion of economists' usual statistical toolbox may be helpful. Statistical Tools for Investigating Sameness and Difference While no set of summary statistics can give a complete picture of the details of similarities and differences between two distributions, the statistics leading to the qualitative judgment of "statistically significant or not" could, for a start, be supplemented by quantitative measures of substantive significance. This section describes two such statistics, which are then, in the next section, used to give an enriched perspective on 28 published articles that address issues of sex and risk. 8 The first measure is Cohen's d, a measure of difference in terms of "effect size" that is already in very wide use in the psychology, education, and neuropsychology literatures. The second is what will be called the Index of Similarity. This measure is newly proposed in the current essay, but is derived from measures already in use in labor and housing research. Cohen's d Cohen's d is one measure of "effect size," expressing the magnitude of a difference between means (e.g., Byrnes, Miller et al. 1999; Wilkinson and Task Force on Statistical Inference 1999; Hyde 2005; Cross, Copping et al. 2011). A difference may most directly, of course, be expressed in the same units as the underlying variable—as a number of dollars, bets, or units on a Likert scale (when such scales are treated as cardinal), etc. But such an expression of difference has two drawbacks. First, as has been stressed in much of the psychology literature, it cannot be easily compared across studies since it is not in standardized units. Secondly, it gives little insight into the substantive significance of the difference. Without knowing how much variation there is within groups, there is no way of knowing whether a between-group difference in means expressed in natural units implies a trivial or a huge divergence in behavior between the groups. Cohen's d goes some way towards relieving these problems by expressing the difference between means in standard deviation units. For the case of a male versus female comparison, it is conventionally calculated as d= where is the male mean, is the female mean, and sp is the pooled standard deviation, a measure of the average within-group variation. 6 As conventionally set up in the psychological literature on gender differences, a positive value for d represents a case where the male score exceeds the female score. The difference is now expressed in standardized (standard deviation) units, and the measure quite sensibly gives a reduced measure of "difference" as the within-group variability (reflected in a rising sp) increases. 6 This is most often estimated as: sp = where sm, sf, nm and nf are the standard deviations and sample sizes for the male and female samples. While this seems to be the most common formula used in the psychology and education literatures, slightly different alternative formulations have also been proposed (e.g., Zakzanis 2001). Econometricians may find an opportunity to make contributions in this area, since some of the existing discussions seem to be weak on statistical theory—for example, Durlak (2009, 923) suggests guidleines that misinterpret the meaning of confidence intervals. 9 Cohen's d can, in theory, take on values from −∞ to +∞, with the extremes occurring the case where all women but no men (or vice versa) share exactly the same particular characteristic (so that the numerator has a nonzero value and there is simultaneously no within-group variation). Perhaps the only variable for which d may be asserted to be infinite—providing one accepts a certain physiological definition of maleness versus femaleness 7—is "Do you have a Y chromosome (1=yes, 0=no)?" This is simply tautological. A Mars-versus-Venus case of disjunctively different "essences" could perhaps also be represented with d-values around 4 or 6, since these are the number of standard deviations that would have to lie between the means of two normal distributions for there to be very little or extremely little overlap between them. In fact, d-values this high do not commonly appear in the literature on sex differences in physiological or behavioral characteristics. One of the largest commonly observable sex differences, for example, is in male and female heights, for which d has been estimated to be about 2.6 (Eliot 2009). 8 Given that heights are approximately normally distributed (and assuming equal variances), Figure 1 gives a picture of roughly how much difference—and how much overlap—is implied between the distribution of women's heights (dashed line) and men's heights (solid line). FIGURE 1 Cohen's d = +2.6 This is clearly a substantial difference—although not Mars-versus-Venus, since we not infrequently observe men and women who are the same height, at heights between the two means. The large d-value does, however, mean that it is relatively rare to observe men who are shorter than the average woman, or women taller than the average man. In cases when (strict) normality can be assumed, d-values can be easily converted into various other measures expressed as percentages of overlap, percentiles, ranks, correlations, or probabilities (Zakzanis 2001; Coe 2002). For example, in the above picture, 99.53% of the men's distribution lies above the female mean. Suppose, instead, that d =.35. Then, again assuming normality and equal variances, the picture would be more like that shown in Figure 2. 7 This definition is, however, disputed by those who identify as transsexual or genderqueer (Factor and Rothblum 2008). 8 Throwing velocity is another characteristic associated with d>+2.0 (Hyde 2005). Presumably these estimates are based on data from the US or other industrialized societies. 10 FIGURE 2 Cohen's d= +.35 Clearly, this difference would be much less observable in everyday life. For example, in the above diagram, a considerably smaller share—64%--of the male distribution lies above the female mean. Whether a given d value is "big," "moderate," or "small" depends a great deal on context and the purpose to which the interpretation of "difference" is being put. Suggested guidelines for qualitative interpretations are readily available in the literature, but should be approached with a great deal of caution. The value of d=+.35, for example, is clearly "small" in the sense that sex would be a quite unreliable signal of, say, above average ability in some skill being measured. To assume that a male advantage at the mean indicates that "men are more able" than women, when d=+.35, would be to ignore the 36% of men who are less able than the average woman, and the 36% of women who are more able than the average man. On the other hand, if being in the upper tail is the basis for employment promotions made on a tournament model, and there is a difference of this size in actual abilities—or merely in employers' perceptions of abilities, as in the case of discriminatory prejudices—d values in the range of low fractions could have a substantial impact (Martell, Lane et al. 1996). The point here is to emphasize that while the language of simple difference tends, given the problems with generics discussed above, to tempt readers into Mars-versusVenus thinking, d-values nuance the discussion of difference by offering one way of quantifying the degree of difference and reminding us of intra-group variability and hence ranges of overlap. Note that d carries no implications about inference to a population. While it is mathematically derived from the same information as inferential statistics such as t- and F- statistics, a large d-value, considered on its own, contains no information relevant to making inferences. Like the t, its numerator is the difference between means, but unlike the t, its denominator is a (weighted) standard deviation, not a standard error. In diametrical contrast, the t statistic contains little information on its own about the substantive magnitude of the difference between the means (other than it is not exactly zero in the sample), since a large t may be in good part due to having a very large sample (and thus a small standard error). Cohen's d, like any statistic, has some drawbacks and hazards in interpretation. It says nothing about differences in variance, skewness, or any other characteristics of the distributions. Its frequent pedagogical presentation in terms of normal distributions with 11 equal variances may lead to a temptation to infer additional characteristics (such as the degree of overlap) even when these conditions do not hold. Note that Cohen's d can be computed—simply as a descriptive statistic—from distributions that do not look at all like the ones pictured above. For example, the d-value for the variable "Never bears a child, 1=true, 0=false" can be crudely estimated to be about +3.02 for adults in the United States. 9 Index of Similarity The Index of Similarity (IS) is an easily computable and understandable measure of overlap that does not rely on an assumption of normality. It can be calculated as IS = Index of Similarity = 1 − 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. The categories may be qualitative (e.g. yes versus no answers), quantitative but limited in number (e.g., the number of lotteries entered out of nine offered), or might be continuous quantitative data aggregated into meaningful groups. IS has an intuitive interpretation as (in equal-sized groups) the proportion of the females and males 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. If IS=.80, for example, it means that 80% of the women could be paired with a man with exactly the same behavior, or vice versa. If one imagines pairing up these matching subjects and setting them aside, it is clear that any differences in the overall distribution—and particularly, any difference in mean scores (when means can be calculated)—must be due to the behavior of remaining 20% of the subjects. IS is hence a direct measure of the overlap of the male and female distributions. IS takes on values from 0 to 1. For the variable "do you have a Y chromosome," IS=0 for males and females if one assumes that chromosomes distinguish male from female. IS=1 for complete matching. IS is unlikely to be zero for non-definitional phenomena, even biologically related ones. For example, about 15% of US women are similar to men, in never bearing children. 10 While one might expect d and IS to be inversely related—so that more "difference" corresponds to less "similarity"—this is not necessarily true, outside of the world of normal distributions with equal variances. For example, if a difference in means is due to a single large outlier, d could be substantial (that is, there is a large difference between the means) while IS exceeds .99 (most subjects are the same). On the other hand, 9 According to U.S. Current Population Survey data from 2008, 17.8% of US women aged 40-44 never had a child (U.S. Census Bureau 2010). Since childbearing after age 40 is still relatively rare, one might guess, conservatively, that the overlap is in the area of around 15%. Thus, for men, the mean is 1 and standard deviation is zero, while for women the mean can be estimated at .15 with an implied standard deviation of .3582. Assuming equal sample sizes, d can be computed as +3.02. 10 See previous footnote. 12 if the shapes or variances of the male and female distributions differ considerably, but in ways that have little overall effect on the means, the result could be a small value for d (little difference in means) and a small value for IS (but relatively few subjects "pair up"). When the underlying distributions are not normal with equal variance, the d and IS measures are complementary, giving two views into a complex reality. Note that, unlike d, IS is non-directional. Knowing that, for example, 80% of the subjects of a study matched exactly does not yield any information about the direction of differences among the remaining 20%. IS is derived from the "index of dissimilarity" (also called "Duncan's D") that has been long used to study racial housing segregation (Duncan and Duncan 1955). The same formula also underlies the "index of occupational segregation" used to study gender segregation of occupations (Reskin 1993; Blau, Ferber et al. 2010, 135). Mathematically, these are the part of the IS equation after the minus sign. They are commonly interpreted as the percent of either group (males or females; blacks or whites) who would have to change their zone of residence (for race) or occupation (for sex) for the responses to be identically distributed across the two groups. As these literatures have pointed out, one problem with such indices is that they are sensitive to the techniques and levels of aggregation used in defining categories (Reskin 1993, 243), so that care must be taken that these are not manipulated to create customized "results." The choice to define an index of similarity in the current essay, instead of dissimilarity, was based on the desire to create a countervailing symmetry with Cohen's d, which measures difference. Magnitudes of Sex Difference and Similarity A study was done of a number of published articles that deal with sex and risk, in order to answer the question, "Given that some studies have found statistically significant differences in measures of risk aversion or risk perception between groups of men and groups of women, what do these results imply about the quantitative magnitudes of the differences between means, and about the overlap of distributions?" Study Design The studies reviewed here were selected in the following manner. We started with those cited in Croson and Gneezy's (2009) meta-analysis, and then added much-cited older articles, did an EconLit search for newer articles, and added other articles as we encountered them in reference lists. The articles are primarily from the fields of economics and finance, though articles from decision science and psychology were also included if they have been cited in the economics literature and/or investigate similar phenomena as the economics literature. While it can be difficult to draw a clear line between "risk" and other behavioral phenomena such as competitiveness or sensation seeking, an effort was made to limit the analysis to studies that examined risk preferences (that is, degrees of willingness to take on risk) and/or risk perceptions (that is, variations in how hazardous a risk is perceived to be). 13 The point of this study being to supplement the usual reports about statistical significance with reports on substantive significance, we sought data from which we could compute d or IS values. In several cases, the necessary information for computing at least some of these statistics was present in the published papers. 11 In two cases we were able to get the necessary information from publically archived supplementary materials or datasets. 12 In addition, a number of authors generously shared their original data or specific statistics with us, on request. 13 In a number of cases, articles contained the results of multiple studies, for only some of which could the necessary information be obtained (either from the article or on request). In other cases, however, and especially with older articles, we were unable to reach authors or authors told us that they could no longer access the relevant datasets. We necessarily, then, could not perform calculations at all for a number of articles—both those showing statistically significant gender differences in the usually expected direction (that is, of lesser male risk aversion or risk perception), as well as articles showing a lack of statistical significance, or statistically significant gender differences in the opposite of the usually expected direction (that is, greater male risk aversion or risk perception, e.g. Shubert, Brown et al. (1999)). 14 As time goes on, we continue to find articles that we would include in the analysis, were we to carry it further. In short, the present study should be interpreted as roughly representative of the literature, rather than as comprehensive. An accurate and complete meta-analysis may not, due to file drawer effects, as well as data availability problems, even be possible. Calculation on a representative group of articles is, however, sufficient for exploring what is meant by "difference." The emphasis in our analysis is on differences between the "raw" distributions of men's and women's performance on various risk-related variables, before adjusting for covariates and, as much as possible, before dividing samples up into subsamples. 15 11 Articles that contained information sufficient to calculate these statistics (or, in some cases in the psychology literature, reported d-values directly) included Arano, Parker et al.(2010), Barsky, Juster et al. (1997), Bernasek and Shwiff (2001), Byrnes, Miller et al. (1999), Carr and Steele (2010), Eriksson and Simpson (2010), (2006), Harris, Jenkins et. al (2006), Lindquist and Säve-Söderbergh (2011), Meier-Pesti and Penz (2008), Olsen and Cox (2001), Powell and Ansic (1997), Rivers, Arvai et al. (2010), Ronay and Kim (2006), Sunden and Surette (1998), and Weaver, Vandello et al. (2012). 12 We appreciate the standards for professional conduct and replication that lay behind the public availability of supplements to Eckel and Grossman (2008) and Holt and Laury (2002). 13 We wish to express our appreciation to the authors of the following articles: Barber and Odean (2001), Beckmann and Menkhoff (2008), Booth and Nolen (2012), Borghans, Golsteyn et al. (2009), Dohmen, Falk et al. (2011) , Eriksson and Simpson (2010), Fehr-Duda, Gennaro et al. (2006), Finucane, Slovic et al. (2000), Gneezy, Leonard et al. (2009), Hartog, Ferrer-i-Carbonell et al. (2002), and Kahan, Braman et al. (2007). 14 Additional studies we reviewed, but which did not result in statistics for Table 1, include Bruhin, FehrDuda et al (2010), Croson and Gneezy (2009), Flynn, Slovic et al. (1994), Levin, Snyder et al. (1988), Olofsson and Rashid (2011), Schubert, Brown et al. (1999), Sunden and Surette (1998), and Tanaka, Camerer et al. (2010). 15 While regression coefficients may also be considered as measures of "effect size," these are not examined here for two reasons. The first is the practical consideration that information on raw distributions is available from, and comparable over, a wider range of published studies. The second reason is that regression results are potentially more influenced by invalid data mining (or “data dredging”) practices. 14