Cumulative standard deviation question

Hello,

I have a question about cumulative standard deviation.

So I am applying this statics equation to my project, which deals with a huge data set. For each second, my data increases so there is no way I can keep track of all the data values.

What I need to find out is the new mean and standard deviation for each second given the previous data (mean, standard deviation, and total number of data), and new data, say x. Of course, each second means I calculate mean and standard deviation for 0,1,2,...second so somewhat accumulating.

So for 1st second, I can calculate new mean using

mean_1 = ((mean_0*n0)+data_1)/(n0+1)

My question is that I am having hard time to find the equation for the new standard deviation, sigma_1 given only mean_0, data_1, n_0, and mean_1

Could anyone help me?

Thank you so much,

Sincerely,