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1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.5. Quantitative Techniques

1.3.5.9.

F-Test for Equality of Two Standard Deviations

Purpose:
Test if standard deviations from two populations are equal
An F-test ( Snedecor and Cochran, 1983) is used to test if the standard deviations of two populations are equal. This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the standard deviations are not equal. The one-tailed version only tests in one direction, that is the standard deviation from the first population is either greater than or less than (but not both) the second population standard deviation . The choice is determined by the problem. For example, if we are testing a new process, we may only be interested in knowing if the new process is less variable than the old process.
Definition The F hypothesis test is defined as:

H0: sigma1 = sigma2
Ha: sigma1 < sigma2     for a lower one tailed test

sigma1 > sigma2     for an upper one tailed test

sigma1 <> sigma2     for a two tailed test

Test
Statistic:
F = s1**2/s2**2

where s1**2 and s2**2 are the sample variances. The more this ratio deviates from 1, the stronger the evidence for unequal population variances.

Significance
Level:
    alpha
Critical
Region:
The hypothesis that the two standard deviations are equal is rejected if

F > F(alpha,N1-1,N2-1)     for an upper one-tailed test

F < F(1-alpha,N1-1,N2-1)     for a lower one-tailed test

F < F(1-alpha/2,N1-1,N2-1)   for a two-tailed test

or

F > F(alpha/2,N1-1,N2-1)

where F(alpha,k-1,N-1) is the critical value of the F distribution with nu1 and nu2 degrees of freedom and a significance level of alpha.

In the above formulas for the critical regions, the Handbook follows the convention that F(alpha) is the upper critical value from the F distribution and f(1-alpha is the lower critical value from the F distribution. Note that this is the opposite of the designation used by some texts and software programs. In particular, Dataplot uses the opposite convention.

Sample Output
Dataplot generated the following output for an F-test from the JAHANMI2.DAT data set:
                       F TEST
 NULL HYPOTHESIS UNDER TEST--SIGMA1 = SIGMA2
 ALTERNATIVE HYPOTHESIS UNDER TEST--SIGMA1 NOT EQUAL SIGMA2
  
 SAMPLE 1:
    NUMBER OF OBSERVATIONS      =      240
    MEAN                        =    688.9987
    STANDARD DEVIATION          =    65.54909
  
 SAMPLE 2:
    NUMBER OF OBSERVATIONS      =      240
    MEAN                        =    611.1559
    STANDARD DEVIATION          =    61.85425
  
 TEST:
    STANDARD DEV. (NUMERATOR)   =    65.54909
    STANDARD DEV. (DENOMINATOR) =    61.85425
    F TEST STATISTIC VALUE      =    1.123037
    DEG. OF FREEDOM (NUMER.)    =    239.0000
    DEG. OF FREEDOM (DENOM.)    =    239.0000
    F TEST STATISTIC CDF VALUE  =    0.814808
  
   NULL          NULL HYPOTHESIS        NULL HYPOTHESIS
   HYPOTHESIS    ACCEPTANCE INTERVAL    CONCLUSION
 SIGMA1 = SIGMA2    (0.000,0.950)         ACCEPT
Interpretation of Sample Output We are testing the hypothesis that the standard deviations for sample one and sample two are equal. The output is divided into four sections.
  1. The first section prints the sample statistics for sample one used in the computation of the F-test.

  2. The second section prints the sample statistics for sample two used in the computation of the F-test.

  3. The third section prints the numerator and denominator standard deviations, the F-test statistic value, the degrees of freedom, and the cumulative distribution function (cdf) value of the F-test statistic. The F-test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance interval printed in section four. The acceptance interval for a two-tailed test is (0,1 - alpha).

  4. The fourth section prints the conclusions for a 95% test since this is the most common case. Results are printed for an upper one-tailed test. The acceptance interval column is stated in terms of the cdf value printed in section three. The last column specifies whether the null hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the F-test statistic cdf value printed in section four. For example, for a significance level of 0.10, the corresponding acceptance interval become (0.000,0.9000).
Output from other statistical software may look somewhat different from the above output.
Questions The F-test can be used to answer the following questions:
  1. Do two samples come from populations with equal standard deviations?
  2. Does a new process, treatment, or test reduce the variability of the current process?
Related Techniques Quantile-Quantile Plot
Bihistogram
Chi-Square Test
Bartlett's Test
Levene Test
Case Study Ceramic strength data.
Software The F-test for equality of two standard deviations is available in many general purpose statistical software programs, including Dataplot.
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