Problem
The standard deviation of the time you take for each statistics homework problem is 1.5 minutes, and it is 2 minutes for each chemistry problem. What is the standard deviation of the time you expect to spend on statistics and physics homework for the week if you have 5 statistics and 4 chemistry homework problems assigned?
Analysis of the Problem
While I haven't learned directly how to calculate combined standard deviation, I have learned about the combined variance formula that looks like this: $$V(aX + bY) = a^2(X) + b^2(Y)$$
Since we're given standard deviation in the problem, I suppose I can calculate the variance and combine the two together. This is what my setup looks like:
$$V(5S + 4C) = 5^2(1.5)^2 + 4^2(2)^2$$
This should give me the output of $$V = 120.25$$
Would it be correct to assume that the combined standard deviation then is the square root of this answer, and thus this?
$$SD = \sqrt{120.25}$$
Best Answer
With one-dimensional variables like this your analysis is correct. The calculated variance is for a new random variable that is the sum of times taken for 5 statistics and 4 chemistry problems, so by definition its standard deviation is the square root of its variance.
However, as the chemistry and statistics answering times are independent, the correct calculation for the variance would be $$5\times1.5^2+4\times2^2=27.25$$