Jan 16

Mean plus deviation-from-mean decomposition

Mean plus deviation-from-mean decomposition

This is about separating the deterministic and random parts of a variable. This topic can be difficult or easy, depending on how you look at it. The right way to think about it is theoretical.

Everything starts with a simple question: What can you do to a random variable X to obtain a new variable, say, Y, whose mean is equal to zero? Intuitively, when you subtract the mean from X, the distribution moves to the left or right, depending on the sign of EX, so that the distribution of Y is centered on zero. One of my students used this intuition to guess that you should subtract the mean: Y=X-EX. The guess should be confirmed by algebra: from this definition


(here we distributed the expectation operator and used the property that the mean of a constant (EX) is that constant). By the way, subtracting the mean from a variable is called centering or demeaning.

If you understand the above, you can represent X as

X = EX+(X-EX).

Here \mu=EX is the mean and u=X-EX is the deviation from the mean. As was shown above, Eu=0. Thus, we obtain the mean plus deviation-from-mean decomposition X=\mu+u. Simple, isn't it? It is so simple, that students don't pay attention to it. In fact, it is omnipresent in Statistics because Var(X)=Var(u). The analysis of Var(X) is reduced to that of Var(u)!

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