Galton’s Bend Revisited: Conditions for Linearity of E[Y|x]

Francis Galton, in his seminal 1886 work titled “Regression Towards Mediocrity in Hereditary Stature", scrutinized the heights of 928 adult offspring along with 205 pairs of parents to elucidate the new concept of “regression toward the mean”. Despite the recurrent use of Galton’s data to exemplify this concept, several studies have contested its conformity to his chosen straight-line linear regression model, attributing a discrepancy to an apparent non-linearity in the data, now recognised as “Galton’s bend". In our paper, we provide a possible explanation for the presence of Galton’s bend by advancing the theory of errors-in-variables regression where we show that E[Y x] E [ Y ∣ x ] is only linear for specific cases of errors-in-variables models.