Something that’s confused me is when to use and when to ignore percentage signs when doing variance ==> std deviation conversions.

The mean amount of rainfall in a city each year is given as 70 inches per year. Standard deviation is 20 inches. Since variance = std devation ^2, then we know that variance = 400, right?

Let’s say a stock’s average rate of return is 7% a year with no dividends. Its standard deviation (omega) is given as 20%. If variance is omega^2, does that mean that variance on the stock’s returns equals

A) 20x20 = 400%?

B) 20% * 20% = 4% or 0.04?

B looks right to me based on equations, but it doesn’t make sense to me because I thought variances were supposed to be greater in value than standard deviations. Which is right?

You are missing the main difference between variance and S.D… Variance doesn’t have any unit whereas S.D. has the same unit in which the given distribution is …

above, u gave example of rainfall in which unit of distibution is inches… So S.D. must be in inches whereas variance is squared S.D. and it doesn’t have any unit… only high number…

Same with your second example where distribution is in percentage… S.D. must also be in percentage but variance must be without any unit/percentage… So both options A) and B) are wrong because both options are in percentages … Option A will be the right one if it’s without percentage sign (400) because variance is σ^2

One of the conceptual problems with variance is that it’s denominated in “units squared”. Since returns in decimal form, variance will be smaller than s.d.

The symbol for standard deviation (of a population) is sigma (σ), not omega (ω).

The correct answer is B) 4% or 0.04.

However, finance types always ignore the percent sign when computing variance, so they always write 400 (but not 400%). Their variances are always to large by a factor of 10,000 (= 100 × 100). I have tried in vain to get them to see the error of their ways, repent, and write variances correctly; it’s a fool’s errand, as it turns out.

So, you have to get accustomed to writing variance = 400. It’s stupid, but it’s the way of the world. Alas.