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.

---

Note

The F distribution is derived from the Student’s t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution. One-Way ANOVA expands the t-test for comparing more than two groups. The scope of that derivation is beyond the level of this course.

---

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.

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

nj = the size of the jth group

sj = the sum of the values in the jth 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

Between group variability:
SS_total = [latex]\displaystyle\sum{{x}^{{2}}}-\frac{{\sum{x}^{{2}}}}{{n}}[/latex]

Total sum of squares:
[latex]\displaystyle\sum{x}^{{2}}-\frac{{{(\sum{x})}^{{2}}}}{{n}}[/latex]

Explained variation: sum of squares representing variation among the different samples:
[latex]\displaystyle{S}{S}_{{\text{between}}}=\sum{[\frac{{({s}{j})}^{{2}}}{{n}_{{j}}}]}-\frac{{(\sum{s}_{{j}})}^{{2}}}{{n}}[/latex]

Unexplained variation: sum of squares representing variation within samples due to chance:
[latex]\displaystyle{S}{S}_{{\text{within}}}={S}{S}_{{\text{total}}}-{S}{S}_{{\text{between}}}[/latex]

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:
[latex]\displaystyle{M}{S}_{{\text{between}}}=\frac{{{S}{S}_{{\text{between}}}}}{{{d}{f}_{{\text{between}}}}}[/latex]

Mean square (variance estimate) that is due to chance (unexplained):
[latex]\displaystyle{M}{S}_{{\text{within}}}=\frac{{{S}{S}_{{\text{within}}}}}{{{d}{f}_{{\text{within}}}}}[/latex]

MS_between and MS_within can be written as follows:

• [latex]\displaystyle{M}{S}_{{\text{between}}}=\frac{{{S}{S}_{{\text{between}}}}}{{{d}{f}_{{\text{between}}}}}=\frac{{{S}{S}_{{\text{between}}}}}{{{k}-{1}}}[/latex]
• [latex]\displaystyle{M}{S}_{{\text{within}}}=\frac{{{S}{S}_{{\text{within}}}}}{{{d}{f}_{{\text{within}}}}}=\frac{{{S}{S}_{{\text{within}}}}}{{{n}-{k}}}[/latex]

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.

---

Note

The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution, because it is assumed that the populations are normal and that they have equal variances.

---

F-Ratio or F Statistic

[latex]\displaystyle{F}=\frac{{{M}{S}_{{\text{between}}}}}{{{M}{S}_{{\text{within}}}}}[/latex]

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

[latex]\displaystyle{F}=\frac{{{n}\cdot{{s}_{\overline{{x}}}^{{ {2}}}}}}{{{{s}_{{\text{pooled}}}^{{2}}}}}[/latex]

where …

• n = the sample size
• df_numerator = k – 1
• df_denominator = n – k
• s²_pooled = the mean of the sample variances (pooled variance)
• [latex]\displaystyle{{s}_{\overline{{x}}}^{{ {2}}}}[/latex] = 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.

 

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)/(k – 1)	F =MS(Factor)/MS(Error)
Error(Within)	SS(Error)	n – k	MS(Error) =SS(Error)/(n –k)
Total	SS(Total)	n – 1

Using a Calculator

One-Way ANOVA Table: The formulas for
SS(Total), SS(Factor) = SS(Between) andSS(Error) = SS(Within) as shown previously.

The 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)/(k– 1)= 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)/(n– k)= 20.8542/7= 2.9792
Total	SS(Total)= 2.2458 + 20.8542= 23.1	n – 1= 10 total data – 1= 9

---

One-Way ANOVA table:

Source of Variation	Sum of Squares ( SS)	Degrees of Freedom ( df)	Mean Square ( MS)	F
Factor (Between)	36,648,561	5 – 1 = 4	[latex]\displaystyle\frac{{{36},{648},{561}}}{{4}}={9},{162},{140}[/latex]	[latex]\displaystyle\frac{{{9},{162},{140}}}{{{2},{044},{672.6}}}={4.4810}[/latex]
Error (Within)	20,446,726	15 – 5 = 10	[latex]\displaystyle\frac{{{20},{446},{726}}}{{10}}={2},{044},{672.6}[/latex]
Total	57,095,287	15 – 1 = 14

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.

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 [latex]\displaystyle\mu=\frac{{{d}{f}{(\text{num})}}}{{{d}{f}{(\text{denom})}}}-{1}[/latex]

References

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

Concept 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

[latex]\displaystyle{S}{S}_{{\text{between}}}=\sum{[\frac{{({s}{j})}^{{2}}}{{n}_{{j}}}]}-\frac{{(\sum{s}_{{j}})}^{{2}}}{{n}}[/latex]

SS_total = [latex]\displaystyle\sum{{x}^{{2}}}-\frac{{\sum{x}^{{2}}}}{{n}}[/latex]

[latex]\displaystyle{S}{S}_{{\text{within}}}={S}{S}_{{\text{total}}}-{S}{S}_{{\text{between}}}[/latex]

df_between = df(num) = k – 1

df_within = df(denom) = n – k

[latex]\displaystyle{M}{S}_{{\text{between}}}=\frac{{{S}{S}_{{\text{between}}}}}{{{d}{f}_{{\text{between}}}}}[/latex]

[latex]\displaystyle{M}{S}_{{\text{within}}}=\frac{{{S}{S}_{{\text{within}}}}}{{{d}{f}_{{\text{within}}}}}[/latex]

[latex]\displaystyle{F}=\frac{{{M}{S}_{{\text{between}}}}}{{{M}{S}_{{\text{within}}}}}[/latex]

F ratio when the groups are the same size: [latex]\displaystyle{F}=\frac{{{n}{{s}_{\overline{{x}}}^{{ {2}}}}}}{{{s}_{{\text{pooled}}}^{{2}}}}[/latex]

Mean of the F distribution:[latex]\displaystyle\mu=\frac{{{d}{f}{(\text{num})}}}{{{d}{f}{(\text{denom})}}}-{1}[/latex]

where:
k = the number of groups n_j = the size of the jth group s_j = the sum of the values in the jth group n = the total number of all values (observations) combined x = one value (one observation) from the data [latex]\displaystyle{{s}_{\overline{{x}}}^{{ {2}}}}[/latex] = the mean of the sample variances (pooled variance)