# 9.6: Chapter 9 Formula Review

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## 9.3 The F-Distribution and the F-Ratio

$M S_{\text {Between }}=\frac{S S_{\text {Between }}}{d f_{\text {Between }}}=\frac{n_{1}\left(\bar{x}_{1}-\bar{x}\right)^{2}+\cdots+n_{g}\left(\bar{x}_{g}-\bar{x}\right)^{2}}{g-1}\nonumber$

$M S_{\text {Within }}=\frac{S S_{\text {Within }}}{d f_{\text {Within }}}=\frac{\left(n_{1}-1\right) s_{1}^{2}+\cdots+\left(n_{g}-1\right) s_{g}^{2}}{n-g}\nonumber$

$F_{obs}=\frac{M S_{\text { between }}}{M S_{\text { within }}}\nonumber$

• $$n_g$$ = the size of the gth group
• $$\bar{x}_g$$ = the mean of the values in the gth group
• $$\bar{x}$$ = the total mean of all observations
• $$g$$ = the number of groups
• $$n$$ = the total number of observations combined
• $$s_g^2$$ = the variance of the values in the gth group

9.6: Chapter 9 Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.