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|Table of Contents Univariate Data Distributions Combination of Several Distributions|
|See also: ANOVA|
For further discussion we have to discriminate two cases: (1) the means of the individual samples are equal, and (2) the means of the individual samples differ from each other (to be specific, the mean of at least one sample is different to the other means).
|The distribution of the combined sample (black) is exactly equal to the distribution of individual samples (the five sample distributions overlap perfectly).|
If the means of the individual samples are equal (according to our assumptions, the variances are equal anyway), the parameters of the combined distributions are the same as the parameters of the idividual ones. The variance of the combined sample is equal to the variances of the individual samples.
|The distribution of the combined sample (black) is considerably broader than the distribution of any of the individual samples. This increase of variance is the basis of the ANOVA (analysis of variance).|
If we combine the individual samples (whose means are not equal) into a larger collective sample the variance of the new sample will be greater than the variance of the individual samples (which is the same for all of them).
This increase in variance of the combined sample gives us now the opportunity to test for the equality of the means of the individual samples, simply by comparing the variances of the individual samples to the variance of the collective sample. Remember, if and only if the means are equal the variance of the collective sample will be the same as the variance of the individual samples. This principle - the testing for the equality of means by investigating the collective variance - is the basis of the ANOVA
Last Update: 2006-Jšn-18