Standard deviation about mean (¯x) for a given discrete frequency distribution x1,x2,x3,.....xn with frequencies f1,f2,f3,...fn is
We know variance for a discrete frequency distribution can be calculated as follows,
σ2 = ∑ni=1fi(xi−¯x)2∑ni=1fi
This can be understood from normal variance calculation as each value xi occurs fi time total number of observations is ∑ni=1fi and each deviation should be added fi times i.e.
→∑ni=1fi(xi−¯x)2
∴ Standard deviation = σ=√∑ni=1fi(xi−¯x)2∑ni=1fi