We now present an application of the bivariate normal distribution.
Example: Heights of parents and children
In the late 19th century, Francis Galton analysed data about the relationship between the heights of children to those of their parents. This was based on measurements of the adult heights of 928 children and the average heights of their parents, with all measurements in inches. (Female heights were multiplied by 1.08 to compensate for sex differences.)
From a jittered scatterplot, both marginal distributions seem reasonably symmetric and normal, so a bivariate normal distribution was fitted to the data, with its parameters estimated by the sample means and variances of the two variables and the sample correlation coefficient.
\[ (C, P) \sim \NormalDistn(\mu_C = 68.09, \mu_P = 68.31, \sigma_C^2 = 6.340, \sigma_P^2 = 3.195, \rho = 0.459 \]Assuming that this bivariate normal distribution is accurate,
(Solved in full version)