Modifying a distribution by adding in samples incrementally
Calculation of standard deviation on the fly is possible (turn to our brothers at math.stackexchange):

https://math.stackexchange.com/questions/198336/how-to-calculate-standard-deviation-with-streaming-inputs

It is easier to keep track of the variance and only take the square root to calculate the stdev when you really need it.

And the mean is even easier, don't overthink it. If you have 4 samples with a mean of 4 and 3 samples with a mean of 3, this totals up to 25 in 7 samples.

So

$$\mu = \frac{\mu_1 N_1 + \mu_2 N_2 + \mu_3 N_3}{N_1 + N_2 + N_3}$$

and

$$\sigma^2 = \frac{\sigma_1^2 N_1 + \sigma_2^2 N_2 + \sigma_3^2 N_3}{(N_1 + N_2 + N_3)^2}$$