Numerator quantifies the difference between the experimental conditions

Denominator is based on a weighted average of the variances within the two groups

You will find that the ANOVA uses the same principle, but expands the number of conditions that can be included in the numerator.

1 2

Pooled

M Mt

SE

From Chocolate Choo-choo data:◦ S2

Auditory = 4.087

◦ S2Image = 1.583

◦ F(1,5) = 4.087/1.583 = 3.987, p>.05◦ C.V. = 6.61

We are safe making the assumption about equal variances in these two samples.

The ratio of two equal variances is 1. Variances cannot be negative so the F-ratios

cannot take on a negative value. In theory, these ratios can take on infinitely

large values, but will only do so on rare occasions.

Like the t-distributions, the F-distributions are really a family of distributions that are determined by the degrees of freedom.

Notice that now we have two degrees of freedom to deal with (numerator and denominator).

At least two independent samples, but typically more than two.

The samples are drawn from approximately normal distributions.

The samples are drawn from populations with equal variances.

Sound familiar?

Within groups variance = Error + Individual Differences

Between groups variance = Error + Individual Differences + ETOH effects

If ETOH effects = 0 then Between/Within = 1 The within groups variance is simply the

average or the variances within each group. We can calculate the Between groups

variance as Σ(MGroup-GM)2/(NGroups-1)◦ We have to multiply this sum by n to convert from

SE2 to S2.

  Group 1 Group 2 Group 3 Group 4

n 25 25 25 25

M 94 101 124 105

s 24 28 31 25

s^2 576 784 961 625

Grand Mean = 106 Variance Between ((94-106)2+(101-106)2+(124-

106)2+(105-106)2)/3 = ??? MSBetween = 164.67*25 = ??? MSWithin = (576+784+961+625)/4 = ??? F = 4116.667/736.5 = ??? dfBetween = 4-1 = 3 dfWithin = 24*4 = 96 C.V.alpha=.05 = 2.70 F(3,96) = 5.590, p < .05