9.6: Chapter 9 Formula Review
- Page ID
- 79082
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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