45 The F Distribution and the F-Ratio

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 number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F_4,10.

To calculate the F ratio, two estimates of the variance are made.

1. Variance between samples: An estimate of σ² that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation.
2. Variance within samples: An estimate of σ² 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. The variance 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 that, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.

MS means “mean square.” MS_between is the variance between groups, and MS_within is the variance within groups.

Calculation of Sum of Squares and Mean Square

• k = the number of different groups
• n_j = the size of the j^th group
• s_j = the sum of the values in the j^th group
• n = total number of all the values combined (total sample size: ∑n_j)
• x = one value:
  ∑x = ∑s_j
• Sum of squares of all values from every group combined:
  ∑x²
• Total sum of squares:
  SS_total =$\sum \left(x^2\right) – \frac{ \left(\sum x\right)^2}{n}$
• Explained variation: sum of squares representing variation among the different samples:
  SS_between = $\sum \left[\frac{(s_j)^2}{n_j} \right] – \frac{\left(\sum s_j\right)^2}{n}$
• Unexplained variation: sum of squares representing variation within samples due to chance:
  SS_within = SS_total – SS_between
• df‘s for different groups (df‘s for the numerator):
  df = k – 1
• Equation for errors within samples (df‘s for the denominator):
  df_within = n – k
• Mean square (variance estimate) explained by the different groups:
  $MS_\text{between}=\frac{{SS}_\text{between}}{{df}_\text{between}}$
• Mean square (variance estimate) that is due to chance (unexplained):
  $MS_\text{within}=\frac{{SS}_\text{within}}{{df}_\text{within}}$

MS_between and MS_within can be written as follows:

• $MS_\text{between}=\frac{{SS}_\text{between}}{{df}_\text{between}}=\frac{{SS}_\text{between}}{k-1}$
• $MS_\text{within}=\frac{{SS}_\text{within}}{{df}_\text{within}}=\frac{{SS}_\text{within}}{n-k}$

The one-way 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.

F-Ratio or F Statistic

$$F=\frac{MS_\text{between}}{MS_\text{within}}$$

If MS_between and MS_within estimate the same value (following the belief that H₀ is true), then the F-ratio should be approximately equal to one. Mostly, just sampling errors would contribute to variations away from one. 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 generally be larger than MS_within.Then the F-ratio will be larger than one. However, if the population effect is small, it is not unlikely that MS_within will be larger in a given sample.

The foregoing calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F-ratio can be written as:

F-Ratio Formula when the groups are the same size

$$F=\frac{n\cdot s_{\bar x}^2}{s_\text{pooled}^2}$$

Data are typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software.

The one-way 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 H₀.