Appendix B. Salquero-Rivera method for calculating a grand standard deviation for a series of measurements each of which has an associated standard error or standard deviation.
Population mean and standard deviation:
We have n samples each with mean value, standard error and sample size.
Equation B.1 relates standard error with standard deviation and sample size n
If we square Eq. B.1, we get Eq. B.2
Furthermore, variance for any sample is defined by Eq. B.3
Replacing B.5 in B.4, we get
Equation B.7 is applied for each sample n
Then, we have for population variance the Eq. B.8
By Eq. B.6, we get population variance in terms of sampling data