Donald Burrill writes: ... > Given a set of numbers X_k (k=1,...n), one calculates a mean by > (1) applying a transformation T to each of the numbers; > (2) summing all the transformed numbers; > (3) dividing that sum by n; > (4) applying the inverse transformation T' to the quotient of (3). ... > This schema has the advantage that the arithmetic mean is more > clearly seen to behave just like the others, with a suitable choice > of transformation (namely, the identity).
Yes, this puts the arithmetic mean neatly in the same form as the others. However, the concept I am after was called `conjugated from arithmetic mean', and saying that the arithmetic mean is conjugated from the arithmetic mean under the identity transform is merely true. It should be possible to "conjugate" from something else. Or if the concept really applies only to the means, then your formulation is more to the point. > In language similar to the notation you used above, we might write > > mean = T'(sum_k (1/n) Txk) > > and the particular mean one gets depends on one's choice of T. Somebody else mailed me and also said that the concept is called a mean (they called it Phi-mean). This is helpful in a way, but does not explain why a mathematician might be happy to hear that this is an instance of a concept called conjugation. Thanks, -- Jussi . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
