\[ F_{(3-1),(N-3)} = \frac{\frac{(n_1\bar{X}^2_1 + n_2\bar{X}^2_2 + n_3\bar{X}^2_3)}{{\sigma^2}}}{\frac{n_1{S}^2_1}{{\sigma^2}} +\frac{n_2{S}^2_2}{{\sigma^2}} + \frac{n_3{S}^2_3}{{\sigma^2}}}\frac{n_1+n_2+n_3 -3}{3-1} \]
dove \(N = n_1 + n_2 + n_3\)