I can't figure out where I have gone wrong here.
There are 3 independent samples all of which have 12 observations, which give values: $\Sigma x_a = 4913, \Sigma x_b = \Sigma x_c =5177, \Sigma x_a^2 = 2094599, \Sigma x_b^2 = 2330695, \Sigma x_c^2 =2257693$.
If i want to calculate $SS_{b}$, I need to first work out each individual sample mean which are $\bar{x_a} = \frac{4913}{12}, \bar{x_b} = \bar{x_c}= \frac{5177}{12} $, and therefore the mean across the whole sample $\bar{x_{t}} = \frac{5089}{12}$. I can then work out $SS_{b} = \Sigma n_i(\bar{x_i} - \bar{x_t}) = 3872.$ According to my calculations.
The mark scheme states an alternative approach when calculating all the values of the ANOVA table. $\Sigma x_t = 15207, \Sigma x_t^2 = 6682987.$ From here this gives $SS_{T} = \Sigma x_t^2 - (\Sigma x_t)^2/36 = 259296.75$, and $SS_{E} = \Sigma x_t^2 - (\Sigma x_i)^2/12 = 83134.917+ 97250.917+ 75718.917 = 256104.75.$ As $SS_T = SS_E +SS_B \Rightarrow SS_B = 3192.$
I can't see where I have gone wrong in my method, any help would be really appreciated.
Thanks.
The solution to your method appears correct, but I think your formula should have been $SS_b=\sum_{i\in{\{a,b,c\}}} n_i(\bar{x_i}-\bar{x_t})^2$
In the answering scheme, $\Sigma x_t$ should be $15267$ instead of $15207$, leading to $SS_T$ being $6682897-((15267^2)/36)$, which is $208506.75$.
Furthermore, $SS_E$ should be as follows (correction highlighted in red):- $SS_E =\sum_{i\in{\{a,b,c\}}}\Sigma x_i^2 - (\Sigma x_i)^2/12 = 83134.917+ 97250.917+ \color{red}{ 24248.917}=204634.75$
Thus $SS_B=SS_T-SS_E=208506.75-204634.75=3872$, which agrees with your calculations.