A Normal Distribution

Many folk who are trans search for some kind of objective legitimation for their sense of gender identity. The internet is awash with tests that promise objective determination of one’s “real” gender. One of the more unlikely such tests is the “digit ratio”: in men and women, the index finger is usually shorter than the ring finger, but an increased difference is associated with high levels of exposure to testosterone in the womb. Thus for males the ratio is 0.947 while for females it is 0.965. On the basis of this simple diagnosis, I’m officially a woman. Or am I?

Measurements are subject to variation, not only because of random errors in measurements (there are limits to the precision of any measurement method) but also because of the intrinsic diversity of biological systems. To understand the meaning of a measurement, therefore, we have to know the spread of measurements. In nature we find that measurements of many properties fit a normal distribution: repeated measurement yields a spread of values that are distributed symmetrically about a mean value. The diagram below shows a normal distribution. The blue vertical line marks the mean, and the two red lines demarcate a region in which 95% of the measurements fall. In statistics, the parameter that is used to quantify the spread of this variation is the standard deviation (SD). The red lines in the diagram lie two standard deviations below and two standard deviations about the mean value.


Armed with this statistical information we can consider again the significance of the digit ratio. For males, the digit ratio has a mean value of 0.951 with a standard deviation 0.035, while for females, the mean value is 0.968 and the standard deviation is 0.028. Thus the mean values lie closer together than one standard deviation. If the sample size is large enough (many thousands of measurements) one might be able to claim that the difference in the values of the digit ratios for males and females is statistically significant, particularly if we compare the standard errors in the means. However, the means are closer together than one standard deviation, and in such circumstances it will be impossible to determine whether a single individual is male or female based on their digit ratio.

One often hears trans folk talking about a “spectrum” of behaviour. “Masculine and feminine are not two entirely different things”, people say; “both genders display a spectrum of behaviour and it is wrong to talk about masculine and feminine attributes”. What they mean is that for each sex there is a spread of behaviours or characteristics. Scientists would expect that such characteristics would fit a normal distribution, but the fact that types of behaviour are distributed normally about a mean does not help us to know whether or not the differences between the sexes are significant.

At the opposite extreme to the digit ratio is the blood testosterone concentration (see figure below). The mean testosterone concentration is very different for men and women: values for the vast majority of the male population lie more than two standard deviations from the female mean value. It should therefore be very easy to make an accurate determination of biological sex based upon a blood test, because statistically speaking there is an enormous difference in the spread of values obtained for the two sexes.


Except that…Philosophers of science like to talk about “ceteris paribus clauses”: they argue that many scientific hypotheses depend upon “all other things being equal”. The importance of this becomes clear when one starts to think about transgender athletes. Martina Navratilova recently provoked a storm of controversy by suggesting that transgender athletes benefitted from having had unusually high levels of testosterone in their bodies for a large part of their lives, giving them an unfair advantage after transition. It’s a very persuasive argument; we know that during the Soviet era, female athletes were treated with testosterone to help them build muscle mass. The competitive advantage that comes from achieving such enhanced testosterone levels is clear; it seems obvious that transgender athletes are enjoying a similar “unnatural” advantage.

These kinds of considerations have led some athletes to suggest that a maximum blood testosterone level be defined in order to provide an objective, non-judgemental demarcation of where an athlete’s blood chemistry can be thought to take them outside the normal range. The marathon runner Paula Radcliffe has lent her support to those arguing for an upper threshold of 5 nMol. Now in fact, even at this level, things are problematic: for a random sample of normal women recently described in one study, several subjects displayed testosterone levels above 5 nMol. However, in sport the problem is rather more complex.

Athletes are not “normal” individuals: they are people who have subjected themselves to strict training regimes to achieve exceptional levels of physical performance. Comparing the blood chemistry of elite athletes with normal distributions collected for entire populations is perhaps problematic. To underline this, one study recently reported blood testosterone concentrations for elite athletes from a wide range of sporting disciplines (for a summary, see this web page). There were two really striking findings. First, a quarter of male athletes had low testosterone concentrations – thus among men testosterone is perhaps less well correlated with athletic prowess than might at first be expected. Second, 5% of female athletes were found to have high levels of testosterone, and a slightly larger number would have failed to qualify under the 5 nMol rule proposed by Paula Radcliffe and others.

The women who displayed very high testosterone concentrations were found to be predominantly competing in track and field and rowing, disciplines where muscle mass is of course important. What these data demonstrate is that a small number of women lie outside the “normal range” and these women are found to be disproportionately significant among elite athletes, because of course athletic competitions are designed to discover and to celebrate exceptional, unusual performance and not average behaviour.

What can we conclude from all of this? It is very hard to draw sharp lines. I do not accept the argument that there exists a “spectrum” of masculine and feminine attributes – I believe that ceteris paribus, we find it quite easy to differentiate between a man and a woman. Genuinely ambiguous individuals are unusual. In all my time mixing with other trans-women, I have rarely been uncertain whether they were really men or women. In a very small number of cases it was genuinely very hard to say, but even after careful application of makeup, its comparatively easy to differentiate between the real McCoy and a complete imposter like myself. This raises many questions about gender identity that are uncomfortable for trans people, who would often prefer that it be mandated that society be unable to make that differentiation for fear of prosecution. In the field of elite athletics, one cannot help but agree with Martina Navratilova that mediocre male athletes may, by virtue of a lifetime of testosterone-assisted development, achieve a degree of athletic prowess relative to natal females after they transition that places them at an unfair advantage and denies female athletes a reward they have worked very hard to achieve. But the normal distribution offers a warning – it’s a great way to think about “normal” but not so smart on “exceptional”!

2 responses to “A Normal Distribution”

  1. If I understood correctly, you are a trans woman who calls herself a complete impostor- so, really a man? If so, what are you going to do? Are you going to revert? Do you use women’s spaces?


  2. I was born male; it is my biology. Long after I am dead, if a forensic anthropologist examines my bones they will conclude that I was a man. However, my femininity is not contrived, nor is it an act; in my feminine presentation I am being true to my feelings.

    As trans people we often inhanbit shades of grey. I feel feminine; but my biology is male. Whatever I feel – and the feelings are very strong – I don’t become a woman simply by changing clothes, although I do acquire an external presentation that is consinant with my inner sense of identity.


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