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.
 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
 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: ∑x^{2}
 Between group variability: SS_{total} = ∑x^{2} – $\frac{\left({\displaystyle \sum {x}^{2}}\right)}{n}$
 Total sum of squares: ∑x^{2} – $\frac{{\left(\sum x\right)}^{2}}{n}$
 Explained variation: sum of squares representing variation among the different samples: SS_{between} = $\sum \left[\frac{{(\text{sj})}^{2}}{{n}_{j}}\right]\frac{{(\sum {s}_{j})}^{2}}{n}$
 Unexplained variation: sum of squares representing variation within samples due to chance: $S{S}_{\text{within}}=S{S}_{\text{total}}\u2013S{S}_{\text{between}}$
 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_{between} = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$
 Mean square (variance estimate) that is due to chance (unexplained): MS_{within} = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$
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.
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
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 
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 $\mu =\frac{df(num)}{df(denom)\u20131}$
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 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 = ________
 Introductory Statistics
 Preface
 Sampling and Data
 Descriptive Statistics
 Introduction
 StemandLeaf Graphs (Stemplots), Line Graphs, and Bar Graphs
 Histograms, Frequency Polygons, and Time Series Graphs
 Measures of the Location of the Data
 Box Plots
 Measures of the Center of the Data
 Skewness and the Mean, Median, and Mode
 Measures of the Spread of the Data
 Descriptive Statistics
 Probability Topics
 Discrete Random Variables
 Introduction
 Probability Distribution Function (PDF) for a Discrete Random Variable
 Mean or Expected Value and Standard Deviation
 Binomial Distribution
 Geometric Distribution
 Hypergeometric Distribution
 Poisson Distribution
 Discrete Distribution (Playing Card Experiment)
 Discrete Distribution (Lucky Dice Experiment)
 Continuous Random Variables
 The Normal Distribution
 The Central Limit Theorem
 Confidence Intervals
 Hypothesis Testing with One Sample
 Hypothesis Testing with Two Samples
 The ChiSquare Distribution
 Linear Regression and Correlation
 F Distribution and OneWay ANOVA
 Appendix A: Review Exercises (Ch 313)
 Appendix B: Practice Tests (14) and Final Exams
 Appendix C: Data Sets
 Appendix D: Group and Partner Projects
 Appendix E: Solution Sheets
 Appendix F: Mathematical Phrases, Symbols, and Formulas
 Appendix G: Notes for the TI83, 83+, 84, 84+ Calculators
 Appendix H: Tables