When a total universe with media T is made of two groups with medias K1 and K2, and population fractions X and X2 respectively, it applies that total media, T = K1*X + K2*(1-X) [1] . This means that if we know 3 of the 4 variables (T,K1,K2,X) we may pinpoint the 4th one exactly.
I think this is an important and simple tool in statistical teaching and theory because it mixes concepts like medias, fractions of population, aggregated coherence, sets theory and practical analysis. I have used it to detect statistical manipulations about income and poverty reports of the government of my country, Colombia (just published).
Do you have other simple answers and comments to this question that I may not have seen in statistical books, papers, and text-books? (Note: Nicholas J. Horton uploaded here in RG: “I hear, I forget. I see, I remember. I do, I understand: a modified Moore-method mathematical statistics course” (September, 24, 2013), a more advanced paper that you may download.)
Emilio,
I think that's a good choice -- it brings in the critical concept of degrees of freedom right away.
I teach statistics to psychology students and I usually use a more abstract starting point: The concept of variance and the goal of accounting for it.
Pat
Thanks Patrick and Alex for your comments which go into the more advanced regions of parametric models that use variance and R tools -like Horton referenced article-. I have in mind fitting models of non-parametric nature for the future. My question refer mainly to high school students first contact with statistics without using combinatory models, but applied to data collected by theirselves using only the media in the first stage, avoiding deviances, p-values, freedom degrees and complex polemic premises at the start. It may be more important to teach them the premises assumed when Laplace criterion is taken -measures are taken as medias of 1/n equal frequence-. With this approach the next step would be to create Lorenz curves for each subset and total, and deeper analysis based in the properties of Lorenz curves. Well, it is just a project in its initial steps. emilio
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