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.
So
Sample 1: |
Sample 2: |
.
.
.
Sample n: |
Where: |
Equation B.1 relates standard error with standard deviation and sample size n
(B.1) |
If we square Eq. B.1, we get Eq. B.2
(B.2) |
Furthermore, variance for any sample is defined by Eq. B.3
(B.3) |
(B.4) |
and
(B.5) |
Replacing B.5 in B.4, we get
(B.6) |
Therefore
(B.7) |
Equation B.7 is applied for each sample n
Then, we have for population variance the Eq. B.8
(B.8) |
By Eq. B.6, we get population variance in terms of sampling data
(B.9) |
where
(B.10) |
and
(B.11) |