How to perform a running (moving) standardization for feature scaling of a growing dataset?
I think you want
$$S_{n}=S_{n−1}+ (x_{n}−μ_{n−1})(x_{n}−μ_{n})$$
where S, x and μ are respectively the variance, value and mean.
See https://fanf2.user.srcf.net/hermes/doc/antiforgery/stats.pdf for explanation and derivation.