Question

What is the Mean Square Error?

Short answer

The Mean square Error is $660.0333$.

Step-by-step solution

Given Information

Group 1	Group 2	Group 3
216	202	170
198	213	165
240	284	182
187	228	197
176	210	201

Explanation

To find the Mean Square Error:

From our Table, we can calculate

$s_1=1017,s_2=1137,s_3=915$

Now, we can calculate SS as below:

$\mathbf{SS}\left(\text{between}\right)=\sum \left[\frac{{\left(s_j\right)}^2}{n_j}\right]-\frac{{\left(\sum s_j\right)}^2}{n}$

$SS\left(\text{between}\right)=\frac{s_1^2}{5}+\frac{s_2^2}{5}+\frac{s_3^2}{5}-\frac{{\left(s_1+s_2+s_3\right)}^2}{15}$

$\text{where}{n}_1=n_2=n_3=5\text{and}n=n_1+n_2+n_3=15$

$\frac{(1017{)}^2}{5}+\frac{(1137{)}^2}{5}+\frac{(915{)}^2}{5}-\frac{(1017+1137+915{)}^2}{15}$

Calculate the sum of squares within groups 

$\mathbf{SS}\left(\text{between}\right)=632856.6-627917.4=\mathbf{4939}\mathbf{.}\mathbf{2}$

$\mathbf{S}\left(\text{total}\right)=\sum x^2-\frac{{\left(\sum x\right)}^2}{n}$

$S\left(\text{total}\right)=640777-627917.4$

$S\left(\text{total}\right)=12859.6$

$\mathbf{SS}\left(\text{within}\right)=\mathbf{SS}\left(\text{total}\right)-\mathbf{SS}\left(\text{between}\right)$

$\text{SS}\left(\text{within}\right)=\mathbf{7920}\mathbf{.}\mathbf{4}$

Calculate the Mean Square

The Mean square Error we calculate like

$M{S}_{\text{within}}=\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}=\frac{S{S}_{\text{within}}}{n-k}$

$M{S}_{\text{within}}=\frac{7920.4}{12}$

$M{S}_{within}=660.03333$