10 11.8 2.4 20 15.3 3.2 15 8.4 4.1

The data is a 3 column numerical data    Column 1 is sample size n    Column 2 is mean value    Column 3 is Standard Deviation value

This page is a simple utility to combine multiple groups of n, mean, and SD into a single group using the following algorithm.

• For each group :
  • Σx = mean * n;
  • Σx² = SD²(n-1)+((Σx)²/n)
• The values are then added together
  • tn = sum of all (n)
  • tx = sum of all Σx
  • txx = sum of all Σx²
• The combine calculations are
  • Combined n = tn
  • Combined mean = tx / tn
  • Combine SD = sqrt((txx-tx²/tn) / (tn-1))
• When calculated the results from the example data should look like the following

  	n	mean	SD		Σx	Σx2
  Grp1	10	11.8	2.4		118	1444.24
  Grp2	20	15.3	3.2		306	4876.36
  Grp3	15	8.4	4.1		126	1293.74
  	tn				tx	txx
  Sum	45				550	7614.34
  Combined	45	12.2222	4.5048

Please Note :

This algorithm must be used with care, as the statistical assumption is that all the groups are merely sub-samples of a single group, and combining them merely restore them back into the original single group.

In many cases this assumption is faulty, as the groups may be from different populations, and sampled under different environments. It is much safer therefore to combine groups using the meta-analysis algorithm, using the Random Effect Model, available in the Meta-analysis for Comparing Two Unpaired Groups Program Page , using the mean and Standard Error of the mean for each group.

The Standard Error of the mean is calculated as SE = SD / sqrt(n) of each group.

After combining them using the Random Effect Model, the Standard Deviation can be recalculated as SD = SE * sqrt(tn), where tn is the sum of sample sizes from all the groups.

References :

This program is a re-engineering of the standard mean and Standard Deviation formula, so there is no reference to this procedure. However, the standard formula for mean and Standard Deviation is available in just about all basic statistical texts. If you must have a reference for it, I used the following

Altman DG, Machin D, Bryant TN and Gardner MJ. (2000) Statistics with Confidence Second Edition. BMJ Books ISBN 0 7279 1375 1. p. 28-31