Cohen describes the effect-size for an independent t-test as d: the difference between means expressed in units of the within-population standard deviation^[1]. The null hypothesis is that that d = 0 .

Cohen set the field's norms by declaring that small-, medium-, and large-effect-sizes are d = 0.20, 0.50, and 0.80 respectively. In other words, "a medium effect" means that the difference between the group-means is half of one standard deviation.

With "a medium effect" of d = 0.50 there is still a lot of overlap between the groups (80%). If you randomly selected one person from each group, there would be a 64% chance that the pair would match the expectation, i.e. that the person from the "higher" group would score higher than the person from the "lower" group (called Probability of Superiority).

Notably, the d for the difference in height between the sexes is about 2.0^[2].
With d = 2.0 there is still some overlap (32%). However, if you randomly selected one male and one female, there would be a 92% chance that the male would be taller.

I mention this effect-size because it helps to remember how huge it is. An effect-size as large as d = 2.0 is the kind of effect you can see just by looking at a room of mixed company. If you see an effect-size reported in a psychology article that is as large or larger than the effect of sex on height, that should set off your skepticism alarms! Most psychological phenomena should not come out as having such a huge effect.

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