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The variance is also called the variation due to error or unexplained variation. When the sample sizes are different, the variance within samples is weighted. Variance within samples-An estimate of σ 2 that is the average of the sample variances, also known as a pooled variance.The variance is also called variation due to treatment or explained variation. If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes.
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Variance between samples-An estimate of σ 2 that is the variance of the sample means multiplied by n, when the sample sizes are the same.To calculate the F ratio, two estimates of the variance are made. It is preferable to use ANOVA when there are more than two groups instead of performing pairwise t-tests because performing multiple tests introduces the likelihood of making a Type 1 error. The scope of that derivation is beyond the level of this course.
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One-way ANOVA expands the t-test for comparing more than two groups. The values of the F distribution are squares of the corresponding values of the t-distribution. The F distribution is derived from the Student’s t-distribution.