I am abstracting data from a research study. I do not know the individual values within each group. But I have 2 groups. One with a population of 149, a mean of 37.3 and an SD of 12.8. My other group has a population of 669, a mean of 38.4 and an SD of 13.4. How do I combine the standard deviations?
[Math] How to combine standard deviations from 2 groups
standard deviationstatistics
Best Answer
Outline of the procedure, based on the so-called 'computational formula' $$S^2 = \frac{\sum_{i=1}^n X_i^2 - n\bar X^2}{n-1}$$ for the sample variance.
$1.$ For Sample 1: Use $n_1,$ $\bar X_1,$ and $S_1^2$ to solve for $Q_1 = \sum_{i=1}^{n_1} X_{1i}^2.$
$2.$ Do the same thing with Sample 2 to solve for $Q_2 = \sum_{i=1}^{n_2} X_{2i}^2.$
$3.$ Then for the combined sample, $Q_c = Q_1 + Q_2.$ Let $n_c = n_1 + n_2.$ Find the combined mean $\bar X_c = \frac{n_1\bar X_1 + n_2\bar X_2}{n_1 + n_2}.$
$4.$ Once again, use the displayed computational formula to 'put the pieces together' to get $S_c^2.$ Take the square root to get $S_c.$