F Distribution and ANOVA: The F Distribution And The F Ratio 1.4 2008/06/23 14:40:52 GMT-5 2008/07/15 11:00:55.434 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org alternate hypothesis ANOVA degrees of freedom F Distribution F Ratio hypothesis test means square null hypothesis One-Way Analysis of Variance population sample Sir Ronald Fisher statistics sum of squares variance This module describes how to calculate the F Ratio and F Distribution based on the hypothesis test for the ANOVA. The distribution used for the hypothesis test is a new one. It is called the F distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator.For example, if F follows an F distribution and the degrees of freedom for the numerator are 4 and the degrees of freedom for the denominator are 10, then F ~ F 4 , 10 .To calculate the F ratio, two estimates of the variance are made.Variance between samples: An estimate of σ 2 that is the variance of the sample means. If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. It is also called variation due to treatment or explained variation. Variance within samples: An estimate of σ 2 that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, the variance within samples is weighted. It is also called the variation due to error or unexplained variation. SS between = the sum of squares that represents the variation among the different samples. SS within = the sum of squares that represents the variation within samples that is due to chance.To find a "sum of squares" means to add together squared quantities which, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Chapter 2. In this very brief overview of ANOVA, we will not go into detail explaining how SS between and SS within are calculated. If you take another statistics course, you will cover sum of squares in detail. Your calculator can easily do the calculations of SS between and SS within . Remember that these two quantities are measures of variability or variation. df between = k - 1 where k is the number of groups (samples). This is the degrees of freedom for the numerator. df within = N - k where N is the total sample size. This is the degrees of freedom for the denominator.MS means "mean square." MS between is the variance between groups and MS within is the variance within groups. MS between = SS between df between = SS between k − 1 MS within = SS within df within = SS within N − k The ANOVA test depends on the fact that MS between can be influenced by population differences among means of the several groups. Since MS within compares values of each group to its own group mean, the fact that group means might be different does not affect MS within .The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MS between and MS within should both estimate the same value.The F-statistic or F-ratio, as it is often called, is F = MS between MS within If MS between and MS within estimate the same value (following the belief that H o is true), then the F-ratio should be approximately equal to 1. Only sampling errors would contribute to variations away from 1. As it turns out, MS between consists of the population variance plus a variance produced from the differences between the samples. MS within is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MS between will be larger than MS within . The F-ratio will be larger than 1.The ANOVA hypothesis test is always right-tailed because larger F-values are way out in the right tail of the F-distribution curve and tend to make us reject H o .

Notation The notation for the F distribution is F ~ F df(num) , df(denom) where df(num) = df between and df(denom) = df within The mean for the F distribution is m = df(num) df(denom) − 1