Variance Properties-Properties of Variance Explained

In this article, we will list the most important properties of variance. The variance is a measure of dispersion that measures the “spread” of the given data values.

1. It is denoted by the symbol \sigma^2. It can be calculated using the formula, \sigma^2 = \frac{\sum_{i=1}^{n} (x_i-\bar{x})^2}{n}.
2. The variance can never be negative. This is clear from the above formula since we are adding the squares of certain quantities. The square of any quantity is always greater then equal to zero. it cannot be negative.
3. The variance is equal to 0 only if the data values are ALL EQUAL to each other. This means that there is no “dispersion” or “spread” in the data. So we see that the variance of a constant series of data is always zero and is strictly positive otherwise.
4. The variance is independent of the “change in origin”. This means that if we subtract a fixed value from a given set of data, the variance not change. Suppose that we are given a set of data values x_1,x_2,\ldots,x_n. Let y_i=x_i-a for some fixed constant a. Then we have that, Var(Y) = Var(X).
5. The variance is affected by the “change in scale”. If we multiply a given set of data by a fixed value, the variance gets multiplied by the square of the scaling factor. Suppose that we are given a set of data values x_1,x_2,\ldots,x_n. Let y_i=cx_i for some fixed constant c. Then we have that, Var(Y) = c^2Var(X).
6. Combining the above two points, we conclude that Var(aX+b)=a^2Var(X) for some fixed constants ‘a’ and ‘b’.
7. If the two variables X and Y are independent then, the variance is additive. This means that, Var(X+Y) = Var(X)+Var(Y) \text{ if X and Y are independent.}
8. If the variables X and Y are not independent the, we obtain a cross-term in the above expression because of the covariance between X and Y. So we have that, Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y) . We see that if X and Y are independent, then Cov(X,Y)=0 and we obtain the previous formula. So we see that this formula generalizes the formula seen in the previous point.
9. The formula for calculating variance in the first point is very cumbersome and tedious for doing calculations. For practical purposes we calculate the variance using the simplified formula, \sigma^2 = \frac{\sum_{i=1}^{n}x_i^2}{n}-(\bar{x})^2.