The Posts of the Great Statisticians: Chi-Square Test for the Variance or Standard Deviation

Chi-Square Test for the Variance or Standard Deviation

When analyzing numerical data, sometimes you need to draw conclusions about the population

variance or standard deviation.

A chi-square test can be used to test if the variance of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test. The two-sided version tests against the alternative that the true variance is either less than or greater than the specified value. The one-sided version only tests in one direction. The choice of a two-sided or one-sided test is determined by the problem. For example:

Finding the Critical Values using the chi-square distribution.

1. For a two-tailed test.

a. Find the degrees of freedom. d.f. = n-1

b. Split the area (α). The area to the right of the larger value CV is α/2 and the area to the right of the smaller value is 1-(α/2)

value at the top of Table G and find the corresponding D.F in the left column.

c. Find the α/2 and 1- (α/2)

d. The critical value is located where the columns meet.

Example: α = 0.02 , d.f. = 7

α/2= 0.01 1- α/2=0.99

C.V. = 1.239

C.V. = 12.017

1. For a Right-tailed test

a. Find the degrees of freedom

b. Find the alpha value at the top of Table G and find the corresponding D.F. in the left column.

c. The critical value is located where the two columns meet.

Example: α = 0.01 , d.f. = 9

C.V. 14.684

1. For a Left-tailed test

a. Find the degrees of freedom

b. Subtract the α value from 1. the left side of the table is used because the chi-square gives the area to the right of the critical value, and the chi square statistic cannot be negative.

c. Find the 1- α value at the top of table G find the corresponding d.f. in the left column.

d. The critical value is located where the two columns meet.