In this post, we discuss the two cases of testing variances. One is testing whether a population variance or standard deviation is equal to some assumed variance. The second case is testing whether two populations have the same variance. By taking samples from the population and computing the sample

variance, we should be able to form decisions whether the hypothesis should be accepted or not.

Case | Null | Test statistic | df |
---|---|---|---|

single sample | $\sigma^2 = \sigma_0^2$ | $\chi^2 =(n-1) \frac{s^2}{\sigma_0^2}$ | $df = n-1$ |

two sample | $\sigma_1^2=\sigma_2^2$ | $f= \frac{s_1^2}{s_2^2}$ | $df1 =n1-1, df2= n2-1$ |

Our file "testofvariances.py" consist of two functions to implement the above cases of testing

variances. For case 1, we call vartest, and for case 2, we call twovartest. Both functions return

a tuple (p-value, test stat, critical value(s)), and both functions allows for three alternatives:left-sided, double sided and right-sided tests.

""" file testofvariances.py author dr.ernesto p. adorio U.P. Clark Clark Field, Pampanga """ from math import * from rstats import * def vartest(svar, pvar0, n, alpha, side=1): """ Performs a basic chisq test for variance with H0: sigma^2 = \sigma_0^2 Arguments: svar - sample variance pvar0 - \sigma_0^2 assumed population variance n - sample size """ teststat = svar/pvar0 * (n -1) df = n-1 if side == 0: crit1 = qchisq(alpha/2, df) crit2 = qchisq(1-alpha/2,df) pvalue = p(teststat, df) if pvalue > 0.5: pvalue = 1-0.5 pvalue *=2 return pvalue, teststat, (crit1, crit2) elif side == -1: crit = qchisq(alpha, df) pvalue = p(teststat, df) return pvalue, teststat, crit else: crit = qchisq(1-alpha, df) pvalue = 1- pchisq(teststat, df) return pvalue, teststat, crit def twovartest(svar1, n1, svar2, n2, alpha, side): """ Performs an F-test for testing the null pvar_1 = pvar2 """ df1 = n1-1 df2 = n2-1 fteststat = float(svar1)/svar2 if side == 0: #print fteststat, df1, df2 crit1 = qf(alpha/2.0, df1, df2) crit2 = qf(1-alpha/2.0, df1, df2) pvalue = pf(fteststat, df1, df2) if pvalue > 0.5: pvalue = 1-0.5 pvalue *=2 return pvalue, fteststat, (crit1, crit2) elif side == -1: crit = qf(alpha, df1, df2 ) pvalue = pf(fteststat, df1, df2) else: crit = qf(1-alpha, df1, df2 ) pvalue = pf(fteststat, df1, df2) return pvalue, fteststat, crit def Walpole_tests(): print "Example 8, p.320" print vartest(1.2**2, 0.9 * 0.9, 10, alpha= 0.05, side=1) print print "Example 9, p.322" print twovartest(16, 12, 25, 10, alpha= 0.10, side = 0) print if __name__=="__main__": Walpole_tests()

When the above file is run under the Python interpreter, it returns the following output:

toto@toto-laptop:~/Blogs/statistics$ python testofvariances.py Example 8, p.320 (0.066881587774126672, 16.0, 16.918977604620448) Example 9, p.322 (0.47864961042191279, 0.64000000000000001, (0.34527730855736571, 3.1024854075283774))

The results looks ok with us at the moment.

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