Addition Rules for Variances

Key Questions

If you add two independent random variables, what is the standard deviation of the combined distribution, if the standard deviations of the two original distributions were, for example, 7 and 5?

#8.60#

You cannot just add the standard deviations.
Instead, you add the variances. Those are built up from the squared differences between every individual value from the mean (the squaring is done to get positive values only, and for other reasons, that I won't delve into).

Standard deviation is defined as the square root of the variance .
The other way around, variance is the square of SD.

So:
- You square the individual SD's to get the variances
- Then you add these together to get the total variance
- Then you take the square root to get the total SD

Your case:
Total variance = #7^2+5^2=49+25=74#
Total SD = #root 2 74=8.60...#

This works for any number of independent variables (mark the bold type for independent!)

MeneerNask · 4 · Jan 24 2015
What is the procedure for calculating the new standard deviation for two combined random variables, if the random variables X and Y are not independent?

Unless you know the "rules" of their dependency, you can't.

MeneerNask · 1 · Mar 5 2015