F Distribution and OneWay ANOVA
The F Distribution and the FRatio
OpenStaxCollege
[latexpage]
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}.
The F distribution is derived from the Student’s tdistribution. The values of the F distribution are squares of the corresponding values of the tdistribution. OneWay ANOVA expands the ttest 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.
 Variance between samples: An estimate of σ^{2} 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.
 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. 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
MS_{between} and MS_{within} can be written as follows:
 \(M{S}_{\text{between}}=\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}=\frac{S{S}_{\text{between}}}{k1}\)
 \(M{S}_{within}=\frac{S{S}_{within}}{d{f}_{within}}=\frac{S{S}_{within}}{nk}\)
The oneway 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 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.
FRatio or F Statistic\(F=\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}\)
If MS_{between} and MS_{within} estimate the same value (following the belief that H_{0} is true), then the Fratio 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 Fratio 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 Fratio can be written as:
FRatio Formula when the groups are the same size\(F=\frac{n\cdot {s}_{\overline{x}}{}^{2}}{{s}^{2}{}_{\text{pooled}}}\)
 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. 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)/(k – 1)  F = MS(Factor)/MS(Error) 
Error
(Within) 
SS(Error)  n – k  MS(Error) = SS(Error)/(n – k)  
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 results are shown in [link].
Plan 1: n_{1} = 4  Plan 2: n_{2} = 3  Plan 3: n_{3} = 3 

5  3.5  8 
4.5  7  4 
4  3.5  
3  4.5 
s_{1} = 16.5, s_{2} =15, s_{3} = 15.7
Following are the calculations needed to fill in the oneway ANOVA table. The table is used to conduct a hypothesis test.
where n_{1} = 4, n_{2} = 3, n_{3} = 3 and n = n_{1} + n_{2} + n_{3} = 10
OneWay ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(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 
As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments
 bare soil
 a commercial ground cover
 black plastic
 straw
 compost
All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the n = 15 plants:
Bare: n_{1} = 3  Ground Cover: n_{2} = 3  Plastic: n_{3} = 3  Straw: n_{4} = 3  Compost: n_{5} = 3 

2,625  5,348  6,583  7,285  6,277 
2,997  5,682  8,560  6,897  7,818 
4,915  5,482  3,830  9,230  8,677 
Create the oneway ANOVA table.
Enter the data into lists L1, L2, L3, L4 and L5. Press STAT and arrow over to TESTS. Arrow down to ANOVA. Press ENTER and enter L1, L2, L3, L4, L5). Press ENTER. The table was filled in with the results from the calculator.
OneWay 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  \(\frac{36,648,561}{4}=9,162,140\)  \(\frac{9,162,140}{2,044,672.6}=4.4810\) 
Error (Within)  20,446,726  15 – 5 = 10  \(\frac{20,446,726}{10}=2,044,672.6\)  
Total  57,095,287  15 – 1 = 14 
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_{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 \(\mu =\frac{df\left(num\right)}{df\left(denom\right)–1}\)
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
\( S{S}_{\text{between}}={\sum }^{\text{}}\left[\frac{{\left({s}_{j}\right)}^{2}}{{n}_{j}}\right]\frac{{\left({\sum }^{\text{}}{s}_{j}\right)}^{2}}{n }\)
\(S{S}_{\text{total}}={\sum }^{\text{}}{x}^{2}\frac{{\left({\sum }^{\text{}}x\right)}^{2}}{n}\)
\(S{S}_{\text{within}}=S{S}_{\text{total}}S{S}_{\text{between}}\)
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 = \(\frac{df\left(num\right)}{df\left(denom\right)1}\)
where:
= 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 oneway ANOVA results are shown in [link].
Group 1  Group 2  Group 3 

216  202  170 
198  213  165 
240  284  182 
187  228  197 
176  210  201 
What is the Sum of Squares Factor?
4,939.2
What is the Sum of Squares Error?
What is the df for the numerator?
2
What is the df for the denominator?
What is the Mean Square Factor?
2,469.6
What is the Mean Square Error?
What is the F statistic?
3.7416
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 oneway ANOVA results are shown in [link].
Team 1  Team 2  Team 3  Team 4 

1  2  0  3 
2  3  1  4 
0  2  1  4 
3  4  0  3 
2  4  0  2 
What is SS_{between}?
What is the df for the numerator?
3
What is MS_{between}?
What is SS_{within}?
13.2
What is the df for the denominator?
What is MS_{within}?
0.825
What is the F statistic?
Judging by the F statistic, do you think it is likely or unlikely that you will reject the null hypothesis?
Because a oneway ANOVA test is always righttailed, a high F statistic corresponds to a low pvalue, so it is likely that we will reject the null hypothesis.
Homework
Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country.
Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.
Northeast  South  West  Central  East  

16.3  16.9  16.4  16.2  17.1  
16.1  16.5  16.5  16.6  17.2  
16.4  16.4  16.6  16.5  16.6  
16.5  16.2  16.1  16.4  16.8  
\(\overline{x}=\)  ________  ________  ________  ________  ________ 
\({s}^{2}=\)  ________  ________  ________  ________  ________ 
H_{0}: µ_{1} = µ_{2} = µ_{3} = µ_{4} = µ_{5}
Hα: At least any two of the group means µ_{1}, µ_{2}, …, µ_{5} are not equal.
degrees of freedom – numerator: df(num) = _________
degrees of freedom – denominator: df(denom) = ________
df(denom) = 15
F statistic = ________