Question

Why is the F-distribution used for the analysis of variance (ANOVA)? What are the characteristics of the F-distribution? How does the F-distribution differ from the z-distribution and the t-distribution?

Answer 1

F test in ANOVA is used for testing equality of means (and various similar analyses). As F- test uses the ratios of variation in between-group divided by within group.

To determine whether the variability between group means is larger than the variability of the observations within the groups.

For ANOVA, you'll reject the hypothesis of equality of means when you get unusually large F-values and you won't reject the hypothesis of equality of means when you get unusually small values

Characteristics of F- Distribution -

1. The F-distribution is positively skewed and with the increase in the degrees of freedom ν1 and ν2, its skewness decreases.
2. The value of the F-distribution is always positive, or zero since the variances are the square of the deviations and hence cannot assume negative values. Its value lies between 0 and ∞.

The difference from other distribution

F distribution is the ratio of two chi-square distributions with degrees of freedom ν1 and ν2, respectively, where each chi-square has first been divided by its degree of freedom. It is positively Skewed and shape is determined by the degrees of freedom.

Z distribution also has bell-shaped and symmetric pdf, but it has lighter tails, which means it tends to produce lesser values that fall far from its mean.

T distribution like the normal distribution is bell-shaped and symmetric, but it has heavier tails, which means it tends to produce values that fall far from its mean