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((Σx)²/n) + (n-1)
• 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))

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.