Understanding Variance and Covariance Concepts in Statistical Analysis

1. Chapter 6 Variance And Covariance

2. Variance And Covariance • Studying sets of numbers as they are is unwieldy. It is usually necessary to reduce the sets in two ways: (1) by calculating averages or measures of central tendency, and (2) by calculating measures of variability

3. Calculation of Means and Variances

4. Kinds of Variance • Population and Sample Variances • Systematic Variance is the variation in measures due to some known or unknown influences that “cause” the scores to lean in one direction more than another. • Between-groups or experimental variance is the variance that reflects systematic differences between groups of measures.

5. Kinds of Variance • Error Variance is the fluctuation or varying of measures that is unaccounted for. Error variance is random variance. • The sampling variance is random or error variance. • However, one should not think that random variance is the only possible source of error variance. Error variance can also consist of measurement errors within the measuring instrument, procedural errors by the researcher, misrecording of responses, and the researcher’s outcome expectancy.

6. Kinds of Variance • A practical definition of error variance would be: Error variance is the variance left over in a set of measures after all known sources of systematic variance have been removed from the measures.

7. An Example of Systematic and Error Variance • P.111-p.114

8. A subtractive Demonstration: Removing Between-Groups Variance from Total Variance • Now assume that, instead of assigning the students to the two groups randomly, we had matched them on intelligence—and intelligence is related to the dependent variable. We now have another source of variance: that due to individual differences in intelligence which is reflected in the rank order of the pairs of criterion measures. • Matching produces systematic variance

9. A recap of Removing Between-Group Variance from Total Variance • In the matching design, there will always be at least two sources of variance. One will be due to systematic sources of variation like individual differences of the subjects whose characteristics or accomplishments have been measured and differences between the groups or subgroups involved in research. The other will be due to chance or random error, fluctuations of measures that cannot currently be accounted for. • Sources of systematic variance tend to make scores lean in one direction or another. This is of course reflected in differences in means.

10. Components of Variance • The case just considered, however, included one experimental component due to the difference between A1 and A2, one component due to individual differences, and a third component due to random error.

11. Covariance • By our previous definition of relation, a set of ordered pairs is a relation. • If we can calculate the variance of any set of scores, is it not possible to calculate the relation between any two sets of scores in a similar way? Is it conceivable that we can calculate the variance of the two sets simultaneously?

12. Covariance • Covxy is an unsatisfactoryindex because its size fluctuates with the ranges and scales of different Xs and Ys. • Product-moment coefficient of correlation is more satisfactory.