Is this adjustment to an rms calculation valid?
You're looking to compute the rolling standard deviation (i.e., a streaming/online algorithm for the standard deviation).  The standard method is to use Welford's method, or a variation.
For the standard deviation, see https://jonisalonen.com/2014/efficient-and-accurate-rolling-standard-deviation/.  This is a minor adjustment to Welford's method (which computes the running variance) to reflect that you want the standard deviation rather than the variance.  See also https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance.