ADDING VARIANCES OR STANDARD DEVIATIONS?

Quite often the students are confronted with the question:  Should we add variances or standard deviations? Let's discuss two specific examples where these two situations could occur.

 Suppose that we know that for the variable X, the mean value and the standard deviation are µX and σX respectively.

a-) In one situation, we are asked to find the mean value and the standard deviation of the variable Y = 2X.

b-) In the other situation, we are asked to find the mean value and the standard deviation of the variable Y defined as the sum of two randomly selected (Xi and Xj) values of X: Yij=( Xi + Xj)

In both cases the mean value of Y is equal µY = 2µX , but what about the standard deviation of Y? Here is where the answers are different.

In the first case (a), we have a linear transformation. For each value of the variable X, we have one value of the variable Y. In another words, the probability of the variable Y (P(Y)) must be equal to the probability of the variable X (P(X)). Therefore,

                          from the definition of Yi = 2Xi, we get:

               

from here, we get that in this case:

             

In the second case(b), we don't have a linear transformation because for each value of Xi, we can have many values of Xj.

Now, the expression :

                    transforms into:

                   

The probability of a specific value of Y; Yij=( Xi + Xj) should be equal to the probability of getting the values  Xi , and Xj : P( Xi and Yj). But these events are independent, so this probability will be equal to the product P(Xi)*P(Yj)

                   

                   

                     

                   

if the variables Xi and Xj are independent, then the third term is equal zero. Observe that this expression is similar to the expression we get for the Linear Correlation Coefficient (r) (see) which is zero when the variables are independent. So,

From the last expression we can see that in this case, the variance ( and not the standard deviation) of Y will be equal to the sum of the variances ( and not the standard deviations) for Xi and Xj.

These results can be extended to the case when we add N independent variables selected randomly from the same population. In this case Y= X1 + X2 +...XN.The mean value for Y will be NµX, and the Variance for Y will be N times the variance for X. This result will be used to demonstrate the Central Limit Theorem (see here!)

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