An assumption of equal variance makes for (relatively) easy & elegant calculations
With regression, for example, one can calculate the ‘best’ straight line which tells us the relationship between 2 variables, a line which we can perhaps only dimly see when the individual values are shown in a scatter plot
Perhaps weight & height
But for regression to be valid one has to assume that the variation of weight in short people is the same as the variation in the tall ones
If this is not tenable, the data are said to be heteroskedastic
All is not lost however. You can try transforming one or both of the variables - perhaps by taking logarithms. Or separating the observations into different groups – say men & women, or adults & children
Quetelet would not have known – formally – about the mathematics of linear regression. This came later, when Francis Galton (Darwins cousin) was trying to work out the relationship between the height of children & their parents & discovered regression to the mean
Quetelet seems to have grasped it instinctively however, when he observed that weight tends to increase according to the square of the height – but only in adults. Children were specifically excluded from this
Links
A treatise on man and the development of his faculties - Google Books Result
Adolphe Quetelet (1796–1874)—the average man and indices of obesity - Garabed Eknoyan
Adolphe Quetelet (1796–1874)—the average man and indices of obesity - Garabed Eknoyan
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