How to calculate standard deviation with streaming inputs? Is there a formula that is capable of operating on streaming inputs and approximating standard  deviation of the set of numbers? 

To add to the accepted answer, a more numerically stable approach (due to Knuth) is to keep track of $\sum (x-\bar{x})^2$ (called M2 in the algorithm below).
The following is copied from the following wikipedia page which is worth a read. https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
def online_variance(data):
    n = 0
    mean = 0.0
    M2 = 0.0

    for x in data:
        n += 1
        delta = x - mean
        mean += delta/n
        M2 += delta*(x - mean)

    if n < 2:
        return float('nan')
    else:
        return M2 / (n - 1)

