We will also standardize all heights to male adult heights -- the approximate way to do this is to add 1" per ft to a womans height -- multiply by 13/12. Let's assume some numbers. In a population, let's assume that male adult height is 70", with a standard deviation of 3". So women have a distribution with mean 64.6" and standard deviation 2.77" (why?).
Now the children will also have Normally distributed heights, depending on
their parents heights, but in order to get a consistent model, we need to
be a bit more detailed. Y will denote a standardized height, and we will
split it into three parts, and overall (constant) mean (70"),
the genetic part X, and an uncorrelated
envoronmental part E:
Y = 70 + X + E
X and E are independent, and the E's of different individuals are
independent. Suppose V(E)=4.5 and V(X)=4.5, so that the standard deviation
of standardized height is 3", as above. Each X and each E has expectation 0.
The X variables are known as the "(additive) genetic values" for height.
Now given a man with genetic value X1, and a woman with genetic value X2,
the genetic value, Xc, of the child is
Xc = (1/2)(X1 + X2) + S
S is another independent Normal random variable. Suppose V(S)=2.25.
The (standardized) height of the child is
Yc = 70 + Xc + Ec
OK, the next bit gets a bit complicated without just a little bit more probability than we have done. I will just tell you, that with these numbers, then a man height Y=(70+ y)" has a genetic value X which is Normal, with mean y/2, and variance 2.25 (St dev 1.5).
So now finally, if parents have heights Y1, Y2*(12/13),
Yc = 70 + (1/2)(X1 + X2) + S + Ec
So, given Y1, Y2, E(Yc) = 70 + (1/4)((y1-70) + (y2-70))
and V(Yc) = (1/4)(2.25+2.25)+ 2.25 + 4.5 = 63/8.
(if I got the arithmetic right this time)
Note that this is less than the population variance which is 9.
The variance of children of parents of given height is going to be
less than the total variation in the population (why?).
Those who have done regression in some class can think about this in those
terms.