Standard deviation will have the same measurement units as the variable itself. Std. deviation of investment returns, it will be dimensioneless (and can be stated in decimal or percents), std. deviation of measurement of your head circumference will be in units of length, etc.
Speaking of dimensional analysis, I always found the variance of a percentage variable to be a strange number. What is the variance of the returns on XYZ security… why, it’s (10%)^2 or 100 (%^2). if 1% * 1% = 1 bps, would that mean that XYZ’s variance is 100 bps? And would that mean that the variance of XYZ is 0.01%? I suppose it would, but I’ve always found % to be an unusual “dimension.” It doesn’t behave the same way as “dollars” or “kilometers” or “hours.” It’s almost as if kilometers = square kilometers.
What do you get if you work with absolute scale instead of multiplying everything by 100 (percent)?
Technically, % is not even a unit. It is merely a convenient conversion factor so we can work with 45% instead of dealing with 0.45. In fact, as a relative measure, since it is always a fraction of one number over a reference multiply by 100, % is really dimension-less. bchadwick, You correctly pointed out the awkwardness of the unit of measurement for variance, and that is one of the reasons why sd is preferred over sd: variance is measured in unit^2, which is less useful than sd, which is in the same unit as the data. But there is no inconsistency between using % (ie 10%) or absolute value (0.1) to measure variance and sd. If you understand % as being merely a conversion factor, then (10%)^2 = (10^2)*(%)^2 = 100*(1/100)^2=1/100=1% You get the same result working with absolute number, (0.1)^2=0.01=1% Remember, % is simply a convenient conversion factor, much like the “kilo” in kilogram.