$$S^2 = \dfrac{N_1 {S_1}^2 + N_2 {S_2}^2 }{ N_1 + N_2 }$$

$$S^2 = \frac{N_1{S_1}^2 + N_2{S_2}^2 +N_1 \left(\bar{x_1} - \bar{x} \right)^2 + N_2 \left( \bar{x_2} - \bar{x} \right)^2 }{N_1 + N_2}$$

หรือ

$$S^2 = \frac{N_1{S_1}^2 + N_2{S_2}^2 +N_1 \left(\bar{x_1}^2 - \bar{x}^2 \right) + N_2 \left( \bar{x_2}^2 - \bar{x}^2 \right) }{ N_1 + N_2 }$$

หรือ 

$$S^2 + \bar{x}^2 = \frac{N_1\left( {S_1}^2 + \bar{x_1}^2 \right) + N_2 \left( {S_2}^2 + \bar{x_2}^2 \right)}{ N_1 + N_2 }$$

$$S^2 = \frac{\displaystyle \sum_{i=1}^k N_i{S_i}^2 + \sum_{i=1}^k N_i \left( \bar{x_i} - \bar{x} \right)^2 }{\displaystyle \sum_{i=1}^k N_i }$$

หรือ

$$S^2 = \frac{\displaystyle \sum_{i=1}^k N_i{S_i}^2 + \sum_{i=1}^k N_i \left( \bar{x_i}^2 - \bar{x}^2 \right) }{\displaystyle \sum_{i=1}^k N_i }$$

หรือ

$$S^2 + \bar{x}^2 = \frac{\displaystyle  \sum_{i=1}^k N_i \left( {S_i}^2 + \bar{x_i}^2 \right) }{\displaystyle \sum_{i=1}^k N_i }$$