  Now in the pure mathematics of the relations between samples and populations, Jacob Bernoulli had in the 18th century established the remarkable fact that, for populations of any large size whatever (say above 2500), the vast majority of samples of 2500 or more closely match in complexion the population from which they are drawn. In the case of our penguins, for example, if the population contains 35% aged 7 years or more, well over nine tenths of samples of 2500 penguins will contain between 33% and 37% aged 7 years or more. This is a necessary, purely mathematical, fact. In the language of statistics, the vast majority of reasonably sized samples are representative of their population, that is, closely resemble it in complexion.