If the average weight of $6$ students is $50\text{kg}$; that of $2$ students is $53\text{kg}$; and that of $2$ students is $55\text{kg}$; then the average weight of all students is:

Total number of students = $6+2+2=10$
Total weight of $6$ students = $6\times 50=300\text{kg}$
Total weight of $2$ students = $2\times 53=106\text{kg}$
Total weight of $2$ students = $2\times 55=110\text{kg}$

Sum of weights of all $10$ students:

$$\text{Total Weight}=300+106+110=516\text{kg}$$

Average weight of all students:

$$\text{Average Weight}=\frac{516}{10}=51.6\text{kg}$$