Q 437. What are the various tests to compare variances for more than 2 populations? How do we use the concept of confidence intervals in these test?

 

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To compare 1-1 variance, we use F-Test or Levene’s Test. To compare variances for 2 or more populations, we use Levene's test or Bartlett's test. If we are sure data are normally distributed, Bartlett's test might have greater statistical power. The hypothesis testing is performed to assess variance of at least 1 group is significantly different from the rest or if all have equal variance.

The individual confidence level indicates how confident we can be that an individual confidence interval contains the population variance of that specific group.

However, because there are multiple confidence intervals in the set, you can be only 95% confident that all the intervals contain the true values.

Each confidence interval may be a range of likely values for Standard deviation of the corresponding population. To maintain the simultaneous confidence level, the Bonferroni confidence intervals are adjusted.

Controlling the simultaneous confidence level is particularly important when multiple confidence intervals are assessed. If the simultaneous confidence level is not controlled, the chance that at least one confidence interval does not contain the true standard deviation increases with the number of confidence intervals.

There 2 tests available to compare variances for more than 2 populations, namely:

 

• Bartlett's test of variance – You can use this test If your sample comes from normally distributed data. In other words, It is used to test the variances are equal for all the samples drawn. It checks that the assumption of equal variances is true before using certain statistical tests like the One-Way ANOVA etc.,. It’s used only when you’re fairly certain that your data comes from a normal distribution. 
• Levene's test – This is an alternate to Bartlett’s test when the sample comes from non-normal data

 

Concept of confidence intervals (CI) :

 

The population variance gives an indication of how the data set is  spread out and  it is typically impossible to know exact population parameter(variance).for this reason, we use a topic in statistics called confidence intervals.. It refers that the probability that a population parameter will fall between a set of values for a certain proportion of times.

For a variance from a normal distribution with unknown mean, a two-sided, 100(1 - α)% confidence interval is calculated and for two-sided intervals, the distance from the variance to each of the limits is different. Thus, instead of stating the distance to the limits we state the width of the interval

There are multiple statistical test are available to compare the variance of different populations. Below are some of the test:

 

F test: This is suitable in case the populations are normally distributed or near to that. This is a parametric test and used for comparing 2 populations variance.

 

Bartlett's test : Like F test , this is also used if the populations are normally distributed or near to that. This is also a parametric test and can be used for comparing more than 2 populations variance.

 

Levene’s Test: This is again a test for variance comparison and can be used for both normal and non-normal data.  This can be used for two or more than two population samples.

 

These test are often used as part of mean comparison test wherever we have assumed equal variance for the population samples. For multiple comparisons in these test , we have individual and Simultaneous Confidence interval.  Individual confidence level will show percentage of confidence that the true population parameter for that population will lie in the Confidence interval. Simultaneous Confidence interval is derived from the individual confidence intervals of multiple populations. It can be interpreted as % of confidence that the entire set of confidence intervals includes the true population standard deviations for all groups.

This question is amongst one of the many misunderstood concepts in Lean Six Sigma. The question particularly asked for tests to compare variances for more than 2 populations, however, there were many responses that described ANOVA and other similar methods. These methods are used to compare the central tendencies (mean or median) of the multiple populations. They do not compare the variances (spread of the data). 

 

From all the approved responses, the best response is from Afzal. Well done!

 

P.S. - responses that detailed ANOVA and other methods have not been approved as they are incorrect responses.