Standard deviation, when to/to not divide by n?

CFA Level I
#1

In the example below were calculating the standard dev of expected returns, and we dont divide by n? Do we only divide by n for populations? or do we only not divide by n for expected returns?

Given the following probability distribution, find the standard deviation of expected returns.

Event P(RA) RA Recession 0.1 -5% Below Average 0.3 -2% Normal 0.5 10% Boom 0.1 31%

Answer:

Find the weighted average return (0.10)(−5) + (0.30)(−2) + (0.50)(10) + (0.10)(31) = 7%.

Next, take differences, square them, multiply by the probability of the event and add them up. That is the variance. Take the square root of the variance for Std. Dev. (0.1)(−5 − 7)2 + (0.3)(−2 − 7)2 + (0.5)(10 − 7)2 + (0.1)(31 − 7)2 = 100.8 = variance.

100.80.5 = 10.04%.

#2

When they give you weights (or probabilities), they’ve already divided by n for you.

Think of it this way: there are 100 (= n) possible future scenarios:

  • in 10 of them, the economy is in a recession, and the return is -5%
  • in 30 of them, the growth of the economy is below average, and the return is -2%
  • in 50 of them, the growth of the economy is normal, and the return is 10%
  • in 10 of them, the economy is booming, and the return is 31%

So, when you multiply -5% by 0.1, what you’re really doing is adding -5% + -5% + -5% + -5% + -5% + -5% + -5% + -5% + -5% + -5% (= -50%), then dividing by 100 (= n).

So . . . whenever they give you weights or probabilities, you just multiply each outcome by its weight or probability, and you’re done. (One caveat: if the weights don’t add up to 1.0, then you have to divide by the sum of the weights. That shouldn’t occur on the exam.)

#3

Fantasitc resposnse. Much appreciated!

#4

My pleasure.

#5

The quick and dirty rule - if they give you weights or probabilities (the same thing, since probabilities are also weights), you don’t divide by “n”. If they don’t, you’re dividing by N (in the case of averages) or N or N-1 (in the case of variances - N for population variance, N-1 for sample variance.

The “regular” average (dividing by N) is also a weighted average - it’s just that all the weights are equal (and all are equal to “1/(N-1)”)