Suppose that X,X1, X2, X3, . . ., Xn are independent indetically distributed random variables, each with Std. Dev. equal to S.
Std. Dev. of nX is equal to nS, but Std. Dev. of (X1 + X2 + . . . + Xn) is equal to Std. Dev. of (X + X + . . . + X) which is equal to sqrt(n)S which is NOT equal to nS.

If the X's are algebraic quantities, then (X1 + X2 + . . . + Xn) = (X + X + . . . + X) = nX. Does the fact that the X's are random variables, rather than algebraic quantities make (X1 + X2 + . . . + Xn) UNEQUAL to (X + X + . . . + X) = nX, and give the sum of n X's and the product nX DIFFERENT Std. Devs? Please explain.

Is (X + X + . . . + X) or nX the appropriate model for the result of n identical bets in a game based on independent trials? Please explain.