1. Exploratory Data Analysis1.3. EDA Techniques1.3.5. Quantitative Techniques

Purpose:Test if thevariance isequal to aspecifiedvalue

A chi-square test ( Snedecor and Cochran, 1983) can be used to testif the variance of a population is equal to a specified value. This testcan be either a two-sided test or a one-sided test. The two-sidedversion tests against the alternative that the true variance is eitherless than or greater than the specified value. The one-sided versiononly tests in one direction. The choice of a two-sided or one-sidedtest is determined by the problem. For example, if we are testing anew process, we may only be concerned if its variability is greaterthan 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

TestStatistic:

where N is the sample size and s is the samplestandard deviation. The key element of this formula isthe ratio s/σ0 which compares the ratio of the samplestandard deviation to the target standard deviation.The more this ratio deviates from 1, the more likelywe are to reject the null hypothesis.

SignificanceLevel:

α.

CriticalRegion:

Reject the null hypothesis that the variance is aspecified value, σ0

2, if

for an upper one-tailedalternative

for a lower one-tailed alternative

or

for a two-tailed alternative

where is the critical value of the chi-square

=2 20

<2 20

>2 20

2 20

T = (N − 1)(s/ 0)2

T > 21− ,N −1

T < 2,N −1

T < 2/2,N −1

T > 21− /2,N −1

2.,N −1

1.3.5.8. Chi-Square Test for the Variance http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm

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distribution with N - 1 degrees of freedom.

The formula for the hypothesis test can easily be converted to forman interval estimate for the variance:

A confidence interval for the standard deviation is computed bytaking the square root of the upper and lower limits of theconfidence interval for the variance.

Chi-SquareTestExample

A chi-square test was performed for the GEAR.DAT data set. Theobserved variance for the 100 measurements of gear diameter is0.00003969 (the standard deviation is 0.0063). We will test the nullhypothesis that the true variance is equal to 0.01.

H0: σ2 = 0.01Ha: σ2 ≠ 0.01

Test statistic: T = 0.3903Degrees of freedom: N - 1 = 99Significance level: α = 0.05Critical values: Χ 2α/2,N-1 = 73.361

Χ 21-α/2,N-1 = 128.422Critical region: Reject H0 if T < 73.361 or T > 128.422

The test statistic value of 0.3903 is much smaller than the lowercritical value, so we reject the null hypothesis and conclude that thevariance is not equal to 0.01.

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

Is the variance equal to some pre-determined threshold value?1.Is the variance greater than some pre-determined thresholdvalue?

2.

Is the variance less than some pre-determined thresholdvalue?

3.

RelatedTechniques

F TestBartlett TestLevene Test

Software The chi-square test for the variance is available in many generalpurpose statistical software programs. Both Dataplot code and Rcode can be used to generate the analyses in this section.

(N − 1)s2

21− /2,N −1

− −−−−−−−−(N − 1)s2

2/2,N −1

− −−−−−−−−

1.3.5.8. Chi-Square Test for the Variance http://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm

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