The variance of a population can be estimated using the chi-square variate, as explained in previous tutorials. Unlike the and distributions, the value of the chi-square variate is defined only for positive values. At a level of significance the or are those values of the variate which give an area in the right tail. Also or are the values of the variate which give an area in the left tail. Similarly is the value of the variate which gives an area in the right tail of the chi-square distribution (curve). However, the value of chi-square which gives an area in the left tail is denoted by , because chi-square cannot be negative.
As an example, at a level of significance of the and will contain 90 percent of the area. Similarly, and will contain 95 percent of the area. We can, therefore state as follows:
Replacing in the middle of the above statement, we get
Dividing both sides of the inequality by , we have
Now, invert the whole inequality, the inequality signs would be reversed, i.e.,
Alternatively, we can also write
Thus, the percent lower and upper confidence limits of the population variances are
Here are the degrees of freedom and the values of
are obtainable from the chi-square table against degrees of freedom and the appropriate level of significance. Also, is the sample variance given by the formula i.e., .
A short cut form of the same formula may be stated as
Example: A random sample of 9 individuals measured 62, 63, 65, 61, 65, 64, 66, 67 and 63 inches in height. Construct a 95 percent confidence interval estimate for the population variance.
Solution: Given that
Using the short cut formula for , we have
Also, consulting the chi-square table against 8 degrees of freedom,
The lower limit of the interval
The upper limit of the interval
Thus, the required interval estimate is