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

1.3.5.8.

Chi-Square Test for the Standard Deviation

Purpose:
Test if standard deviation is equal to a specified value

A chi-square test ( Snedecor and Cochran, 1983) can be used to test if the standard deviation 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 standard deviation 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, if we are testing a new process, we may only be concerned if its variability is greater than the variability of the current process.

Definition

The chi-square hypothesis test is defined as:

H0:
Ha:	for a lower one-tailed test       for an upper one-tailed test       for a two-tailed test
Test Statistic:	T = where N is the sample size and is the sample standard deviation. The key element of this formula is the ratio which compares the ratio of the sample standard deviation to the target standard deviation. The more this ratio deviates from 1, the more likely we are to reject the null hypothesis.
Significance Level:	.
Critical Region:	Reject the null hypothesis that the standard deviation is a specified value, , if        for an upper one-tailed alternative     for a lower one-tailed alternative   for a two-tailed test or where is the critical value of the chi-square distribution with N - 1 degrees of freedom. In the above formulas for the critical regions, the Handbook follows the convention that is the upper critical value from the chi-square distribution and is the lower critical value from the chi-square distribution. Note that this is the opposite of some texts and software programs. In particular, Dataplot uses the opposite convention.

The formula for the hypothesis test can easily be converted to form an interval estimate for the standard deviation:

Sample Output

Dataplot generated the following output for a chi-square test from the GEAR.DAT data set:

                   CHI-SQUARED TEST
                 SIGMA0 =   0.1000000
 NULL HYPOTHESIS UNDER TEST--STANDARD DEVIATION SIGMA = .1000000
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      100
    MEAN                        =   0.9976400
    STANDARD DEVIATION S        =   0.6278908E-02
  
 TEST:
    S/SIGMA0                    =   0.6278908E-01
    CHI-SQUARED STATISTIC       =   0.3903044
    DEGREES OF FREEDOM          =    99.00000
    CHI-SQUARED CDF VALUE       =    0.000000
  
                     ALTERNATIVE-          ALTERNATIVE-
    ALTERNATIVE-     HYPOTHESIS            HYPOTHESIS
    HYPOTHESIS       ACCEPTANCE INTERVAL   CONCLUSION
 SIGMA <> .1000000    (0,0.025), (0.975,1)  ACCEPT
 SIGMA <  .1000000    (0,0.05)              ACCEPT
 SIGMA >  .1000000    (0.95,1)              REJECT

Interpretation of Sample Output

We are testing the hypothesis that the population standard deviation is 0.1. The output is divided into three sections.

1. The first section prints the sample statistics used in the computation of the chi-square test.

2. The second section prints the chi-square test statistic value, the degrees of freedom, and the cumulative distribution function (cdf) value of the chi-square test statistic. The chi-square test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section three. For an upper one-tailed test, the alternative hypothesis acceptance interval is (1 - ,1), the alternative hypothesis acceptance interval for a lower one-tailed test is (0,), and the alternative hypothesis acceptance interval for a two-tailed test is (1 - /2,1) or (0,/2). Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis.

3. The third section prints the conclusions for a 95% test since this is the most common case. Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section two. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the chi-square test statistic cdf value printed in section two. For example, for a significance level of 0.10, the corresponding alternative hypothesis acceptance intervals are (0,0.05) and (0.95,1), (0, 0.10), and (0.90,1).

Output from other statistical software may look somewhat different from the above output.

Questions

The chi-square test can be used to answer the following questions:

1. Is the standard deviation equal to some pre-determined threshold value?
2. Is the standard deviation greater than some pre-determined threshold value?
3. Is the standard deviation less than some pre-determined threshold value?

Related Techniques

F Test
Bartlett Test
Levene Test

Software

The chi-square test for the standard deviation is available in many general purpose statistical software programs, including Dataplot.