Here are the table of values: $$ \begin{array} 1 x_j&n_j\\ 450.5&4\\ 510.5&8\\ 570.5&12\\ 630.5&6 \end{array}$$
$$N=30$$
$$\bar x=550.5$$
$$S_x=\sqrt\frac{\sum_{j=2}^4n_jx_j^2-N\bar x}{N-1}=57.53$$
my explanation $S_X=\sqrt\frac{(4\times450.5^2+8\times510.5^2+6\times630.5^2+12\times570.5^2)-(30\times4\times550.5)}{29}=377.4281389$
solution=$S_x$= 57.53 but my explanation $S_x$=377.428.
where is my mistake??
please teach me
I suppose you simply have the wrong formula: You are missing a square on the $\bar x$ and it should read $$S_x=\sqrt\frac{\sum_{j=1}^4n_jx_j^2-N\bar x^2}{N-1}.$$ Using this instead of the one you wrote, Wolframalpha confirms the result.