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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 forthe 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.
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 andthe sample standard deviation in Descriptive Statistics .
$\mathrm{MS}$ means "mean square." ${\mathrm{MS}}_{\text{between}}$ is the variance between groups and ${\mathrm{MS}}_{\text{within}}$ is the variance within groups.
${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ can be written as follows:
The OneWay ANOVA test depends on the fact that ${\mathrm{MS}}_{\text{between}}$ can be influenced by population differences among means of the several groups. Since ${\mathrm{MS}}_{\text{within}}$ compares values of each group to its own group mean, the fact that group means might be different doesnot affect ${\mathrm{MS}}_{\text{within}}$ .
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 samplegroups come from populations with different normal distributions. If the null hypothesis is true, ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ should both estimate the same value.
If ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ estimate the same value (following the belief that ${H}_{o}$ is true), then the Fratio should be approximately equal to 1. Mostly just sampling errorswould contribute to variations away from 1. As it turns out, ${\mathrm{MS}}_{\text{between}}$ consists of the population variance plus a variance produced from the differences between thesamples. ${\mathrm{MS}}_{\text{within}}$ is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, ${\mathrm{MS}}_{\text{between}}$ will generally be larger than ${\mathrm{MS}}_{\text{within}}$ . Then the Fratio will be larger than 1.However, if the population effect size is small it is not unlikely that ${\mathrm{MS}}_{\text{within}}$ will be larger in a give sample.
The above 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:
The data is typically put into a table for easy viewing. OneWay ANOVA results are often displayed in this manner by computer software.
Source of Variation  Sum of Squares (SS)  Degrees of Freedom (df)  Mean Square (MS)  F 

Factor
(Between) 
SS(Factor)  k  1  MS(Factor) = SS(Factor)/(k1)  F = MS(Factor)/MS(Error) 
Error
(Within) 
SS(Error)  n  k  MS(Error) = SS(Error)/(nk)  
Total  SS(Total)  n  1 
Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The OneWay ANOVA table is shown below.
Plan 1  Plan 2  Plan 3 

5  3.5  8 
4.5  7  4 
4  3.5  
3  4.5 
OneWay ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) are shown above. This same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively).
Source of Variation  Sum of Squares (SS)  Degrees of Freedom (df)  Mean Square (MS)  F 

Factor
(Between) 
SS(Factor)
= SS(Between) =2.2458 
k  1
= 3 groups  1 = 2 
MS(Factor)
= SS(Factor)/(k1) = 2.2458/2 = 1.1229 
F =
MS(Factor)/MS(Error) = 1.1229/2.9792 = 0.3769 
Error
(Within) 
SS(Error)
= SS(Within) = 20.8542 
n  k
= 10 total data  3 groups = 7 
MS(Error)
= SS(Error)/(nk) = 20.8542/7 = 2.9792 

Total  SS(Total)
= 2.9792 + 20.8542 =23.1 
n  1
= 10 total data  1 = 9 
The OneWay ANOVA hypothesis test is always righttailed because larger Fvalues are way out in the right tail of the Fdistribution curve and tend to make us reject ${H}_{o}$ .
The notation for the F distribution is $F$ ~ ${F}_{\text{df(num)},\text{df(denom)}}$
where $\text{df(num)}={\mathrm{df}}_{\text{between}}$ and $\text{df(denom)}={\mathrm{df}}_{\text{within}}$
The mean for the F distribution is $\mu =\frac{\mathrm{df(num)}}{\mathrm{df(denom)}1}$
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