7
Jan 16

## 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

$EY=E(X-EX)=EX-E(EX)=EX-EX=0$

(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 $Var(u)$!

### 5 Responses for "Mean plus deviation-from-mean decomposition"

1. […] a distribution is as important as centering or demeaning considered here. The question we want to find an answer for is this: What can you do to a random variable to […]

2. […] we know that that can be achieved by centering and scaling. Combining these two transformations, we obtain the definition of the z […]

3. […] (5) is the mean-plus-deviation-from-the-mean decomposition. In more complex situations, deriving its analog greatly simplifies the proof. Many students think […]

4. […] cases. a) Letting in the linearity property we get . This is called additivity. See an application. b) Letting in (1) we get . This property is called homogeneity of degree 1 (you can pull the […]

5. […] means of the errors are the same:  for all and 2) the means are different. Read the post about centering and see if you can come up with the answer for the first question. I'm afraid there is nothing you […]