I was chatting with a friend about a few quantitative genetic “back-of-the-envelopes.” Specifically, about the expectation of the heights of the offspring of any given couple in the United States. I say the United States because it is a nation where most people get enough to eat; that means that heritability is on the order of 80-90% for this trait. By this, I mean that 80-90% of the variation in height we see within the population is due to variation in genetics. Those who are tall are likely to have tall parents, and those who are short are likely to have short parents. The key is likely of course, expectation is not a guarantee.
Looking around there’s a lot of numbers about the mean heights of white males & females in the United States. I’ll use 5 feet 10 inches for men and 5 feet 4 inches for women. That’s 70 inches and 64 inches respectively. It might be off by an inch or so, but that’s about it (if you care, black Americans are about the same height, though Asian and Mexican Americans are shorter). Let’s take the standard deviation to be 3 inches (most of the numbers are around this value, if a touch smaller).
So what happens when you have a husband and wife pair where the husband is 6 feet 1 inches tall and the wife 5 feet and 6 and 1/2 inches tall? In other words, 73 inches for the male and 66.5 inches for the female. That translates into:
1 standard deviation above the norm for the male
0.5 standard deviations above the norm for the female
OK, what does that tell us about the offspring? Well, height is a quantitative trait; it’s due to the combined action of hundreds or thousands of genes. You can basically pretend as if blending is operative because the discrete units are so small that they converge onto a continuous distribution. With a trait like height you can take the mid-parent value and then correct for its heritability, that is, the proportion of the trait which is controlled by genetic variation. So for the parents above the expectation for the offspring would be 0.75 standard deviations above the norm, 2.25 inches. But height is only 80-90% heritable, the remainder is environmental variance is which isn’t heritable, it won’t transmit from parent to offspring. So you need to correct for it. Let’s take heritability as 80%, so that 1/5 of the variance is environmental and will reflect the population distribution. In other words, the offspring expectation will regress 1/5 back toward the population mean, so:
The offspring is expected to be 1.8 inches above the mean value for their sex, 65.8 inches (nearly 5 feet 6 sinches) for a female and 71.8 inches for a male (nearly 6 feet).
There is variance around the mean. Height is normally distributed, and you can assume that the distribution of the offspring will be normal as well about the new mean. So for the distribution of male offspring the mean is 71.8 inches with a standard deviation of 3 inches and for females 55.8 inches with a standard deviation of 3 inches. Concretely, for female offspring 70% will fall within the interval of 62.8 inches to 68.8 inches. For males the equivalent interval will be 68.8 inches and 74.8 inches.
I rigged the numbers to be somewhat simple, but the point is that you can perform these sorts of back-of-the-envelopes without paper & pencil. Basic human genetics doesn’t need to be arcane or astrological, it’s deducible from some spare first principles. And as you can see above, it can also illustrate how evolution works upon quantiative traits.