12.2: Chapter 12 Formulas
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 12 Formulas
Between-Group Variance
The between-group variance in ANOVA, denoted as \(S_B^2\), measures the variability of group means around the overall (grand) mean. It reflects how much the means of different groups differ from one another. A larger between-group variance suggests that the group means are more spread out, which may indicate a significant effect of the independent variable.
\(S_B^2 = \dfrac{\sum n_i (\bar{x}_i - \bar{x}_{\text{GM}})^2}{k - 1}\)
Where:
- \( S_B^2 \) is the between-group variance
- \( n_i \) is the number of observations in group \( i \)
- \( \bar{x}_i \) is the mean of group \( i \)
- \( \bar{x}_{\text{GM}} \) is the grand mean (mean of all data points)
- \( k \) is the number of groups
Within-Group Variance
The within-group variance in ANOVA, denoted as \(S_W^2\), estimates the average variability within each group by pooling the individual group variances, weighted by their degrees of freedom. This form emphasizes that the total degrees of freedom within groups is the sum of \(n_i - 1\) across all groups.
\(S_W^2 = \dfrac{\sum (n_i - 1) s_i^2}{\sum (n_i - 1)}\)
Where:
- \( S_W^2 \) is the within-group variance
- \( s_i^2 \) is the variance of group \( i \)
- \( n_i \) is the number of observations in group \( i \)
- \( \sum (n_i - 1) s_i^2 \) is the weighted sum of variances from each group
- \( \sum (n_i - 1) \) is the total degrees of freedom within groups
Test Statistic For ANOVA
The test statistic for ANOVA is used to determine whether there are statistically significant differences between the means of three or more independent groups. It is calculated as the ratio of the between-group variance to the within-group variance. A larger test statistic value indicates greater variation between the group means relative to the variation within the groups, which may suggest that at least one group mean is different from the others.
\(F = \dfrac{S_B^2}{S_W^2}\)
Where:
- \( F \) is the ANOVA test statistic
- \( S_B^2 \) is the between-group variance
- \( S_W^2 \) is the within-group variance
Authors
"12.2: Chapter 12 Formulas" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY 4.0