13.2 The f distribution and the f-ratio  (Page 2/5)

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

If MS _between and MS _within estimate the same value (following the belief that H ₀ 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

Where ...

• n = the sample size
• df _numerator = k – 1
• df _denominator = n – k
• s ² pooled = the mean of the sample variances (pooled variance)
• $s_{\overline{x}}{}^2$ = the variance of the sample means

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 H ₀ .

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 $\mu =\frac{df(num)}{df(denom)–1}$

References

Tomato Data, Marist College School of Science (unpublished student research)

Chapter review

Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is the F statistic from an F distribution with (number of groups – 1) as the numerator degrees of freedom and (number of observations – number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table.

Formula review

$S{S}_{\text{between}}={{\displaystyle \sum }}^{\text{}}\left[\frac{{(s_j)}^2}{n_j}\right]-\frac{{\left({{\displaystyle \sum }}^{\text{}}{s}_j\right)}^2}{n }$

$S{S}_{\text{total}}={{\displaystyle \sum }}^{\text{}}{x}^2-\frac{{\left({{\displaystyle \sum }}^{\text{}}x\right)}^2}{n}$

$S{S}_{\text{within}}=S{S}_{\text{total}}-S{S}_{\text{between}}$

df _between = df ( num ) = k – 1

df _within = df(denom) = n – k

MS _between = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$

MS _within = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$

F = $\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}$

F ratio when the groups are the same size: F = $\frac{n{s}_{\overline{x}}{}^2}{s^{\text{2}}{}_{pooled}}$

Mean of the F distribution: µ = $\frac{df(num)}{df(denom)-1}$

where:

• k = the number of groups
• n _j = the size of the j ^th group
• s _j = the sum of the values in the j ^th group
• n = the total number of all values (observations) combined
• x = one value (one observation) from the data
• $s_{\overline{x}}{}^2$ = the variance of the sample means
• $s^2{}_{pooled}$ = the mean of the sample variances (pooled variance)

Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in the table are the weights for the different groups. The one-way ANOVA results are shown in [link] .

Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in [link] .