Transform means and variances and combine summaries.
Notes and key methods
If \(Y=aX+b\), then \(E(Y)=aE(X)+b\) and \(\operatorname{Var}(Y)=a^2\operatorname{Var}(X)\). Hence standard deviation scales by |a|; translation changes the mean but not the variance.
To combine groups, recover \(\sum x=n\bar x\) and \(\sum x^2=n(v+\bar x^2)\), add the two totals and divide by the combined number of observations. Averaging two means is valid only when the sample sizes are equal.
Check the conditions
The two groups are disjoint when combining observations.
State whether variances use n or n−1 before recovering sums of squares.
Three worked examples
Example 1 · Coding and combined data
The n=7 observations X have mean 4 and population variance 5. Define \(Y=4X+9\). A second disjoint group has 9 Y-values with mean 29.