Problems with Variance ©2005 Dr. B. C. Paul

Determining What To Do We have looked at techniques that depend on normally distributed data with variance limited to some set of values. We have considered how to tell if a distribution is not normally distributed but not yet what to do  Besides hope for the best if the condition is border-line We have done many tests that assume that all samples have the same variance  T tests, ANOVA, to a lesser extent regression  We don’t yet know how to detect if there is a problem or what to do about it.

Variance Problems with T tests T tests are often used to decide if samples taken from two populations have a different average value. Example- Quincy Quality has a factory that makes boat widgets. Widgets that do not meet the quality control tests at the end have to be discarded and add cost to the operation. Quincy wants to know if his night and day shifts have the same rate of widget rejection so he looks at records of number of widgets rejected versus number produced on each assembly line.

Quincy Enters his Sample Data Quincy selects 20 days at random For his day and night shifts and Looks at the number of rejects per Thousand units manufactured. He enters the data into his trusty SPSS program as shown.

Quincy Does a T test to see if the means of the two shifts are equal Quincy goes to Analyze and clicks to get The pull down menu. He highlights compare means to get the Pop out side menu He highlights and clicks independent Samples T test.

He Sets Rejects as His Test Variable

He Needs to Use a Variable to Break His Data Up by Group Since he is concerned about shift he Enters shift as his grouping variable But the program still wants to know How he will use this variable to break Up his groups.

He Tells the Program Shift 1 and Shift 2 After clicking define groups the Define group menu pops up Quincy enters that group 1 has a Shift value of 1 (Day Shift) and Group 2 has a value of 2 (Night Shift) He will then click continue.

Quincy Checks His Options Quincy clicks options to bring up the Options menu. He has an option to determine what Kind of confidence interval he wants. The default is 95% which sounds good For now. Quincy Clicks continue And then OK.

Oh Hear Come the Results! The Day Shift averages 26.1 Rejects per thousand. The Night Shift averages 33. The samples exhibit a difference of 6.95 Rejects per thousand.

Interpreting The T statistic for this difference is Our confidence that one shift is in fact actually better than The other is 98.7%

Quincy’s Assumptions To Draw the conclusion Quincy must believe his rate of rejects is normally distributed and That the day to day (night to night?) differences in the number of rejects is the same for both samples  But is the variance really the same?

Looking Closer The F test for homogeneity of variances – ie the variances are The same for both shifts – is 25.3 The significance of that statistic is essentially 100% - ie there is no Chance in H_ _ _ that the day to day variance of the shifts is the Same.

How did we do that Thank goodness we didn’t have to (the computer did it for us) A sample variance estimated from samples of a normal distribution follows a Qui Squared distribution. To test variances we divide one variance by the other.  Two Qui Squared distributions – one dividing the other has an F distribution. If the F value is too strange we reject the null hypothesis that our variances are equal.

Ok We Reject it – Now What? The validity of our T test depended on homogeneity variance which we just lost  However Quincy suspects that his night shift might not be doing to well and wants to go kick some – well you get it The solution was put together by Brehens and Fisher.  When variance is not equal you get a slightly distorted T distribution  Rather than come up with a separate distribution table the distorted distribution will pretty well match a T distribution if you fudge the degrees of freedom For us the best thing is the computer does this degree of freedom fudging for us and just splits out the result.

So Does Quincy Get to Kick _ _ _ _ Without assuming homogeneity our T statistic stays the Same but the probability of getting the result by chance rises To 1.6%. 98.4% - Quincy is sure his night shift is messing up.