A commute has two legs; a package's weight is box plus contents; a test score is part 1 plus part 2. Each piece is random, and we want the mean and spread of the total. You already know how shifting and scaling affect the mean and standard deviation of one variable. Now we combine two.
Key idea
For any random variables: $E(X + Y) = E(X) + E(Y)$ and $E(X - Y) = E(X) - E(Y)$. If $X$ and $Y$ are independent: $$\text{Var}(X + Y) = \text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y).$$ Variances add; standard deviations do not. Also $E(a + bX) = a + bE(X)$ and $\sigma_{a + bX} = |b|\sigma_X$.
Why it works
Means add because averages of sums are sums of averages. For variance, write the deviation of the sum as $(X - \mu_X) + (Y - \mu_Y)$ and square it: $$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2E[(X - \mu_X)(Y - \mu_Y)].$$ For independent variables the last term is 0: a high X is equally likely to come with a high or a low Y, so the cross products cancel on average. For $X - Y$, the cross term flips sign but is still 0, and $\text{Var}(-Y) = (-1)^2\text{Var}(Y)$, so the variances add again. Like the legs of a right triangle, independent spreads combine by the Pythagorean rule, not by plain addition.
Worked example: A two-part commute
The drive to the station has mean 20 minutes and SD 3. The train ride has mean 35 minutes and SD 4, independent of the drive. Find the mean and SD of the total time.
- Mean: $20 + 35 = 55$ minutes.
- Variances: $3^2 + 4^2 = 9 + 16 = 25$.
- SD: $\sqrt{25} = 5$ minutes (not $3 + 4 = 7$).
Answer: Mean 55 minutes, SD 5 minutes.
Worked example: A difference
With the same X and Y, find the SD of $Y - X$, how much longer the train ride is than the drive.
- Mean: $35 - 20 = 15$ minutes.
- Variances still add: $\text{Var}(Y - X) = 16 + 9 = 25$.
- SD: $5$ minutes. Subtracting one uncertain amount from another does not cancel the uncertainty.
Answer: Mean 15 minutes, SD 5 minutes.
Worked example: Many independent pieces
Each bag of apples weighs a mean of 10 pounds with SD 2 pounds, independently. Find the mean and SD of the total weight of 4 bags.
- Mean: $4 \times 10 = 40$ pounds.
- Variance: $4 \times 2^2 = 16$.
- SD: $\sqrt{16} = 4$ pounds, which is $2\sqrt{4}$, not $4 \times 2 = 8$.
Answer: Mean 40 pounds, SD 4 pounds.
Common mistake
Adding standard deviations ($3 + 4 = 7$), or subtracting variances for a difference ($16 - 9$). Check: for independent pieces, always square the SDs, add, then take the square root, whether you are adding or subtracting the variables.