Difference between Standard Deviation and Variance.
In: Mathematics
One important characteristic of a set of data is its dispersion– basically, how much its values are spread out. So how do we stick a number to measure the dispersion of a set of data? Lets say that X is my set of data, and μ is my mean. Your first thought might be to calculate the average of the deviations of X from the mean. In other words, you’d calculate the mean of X-μ (if that makes sense). However, this results immediately in a problem. The negative deviations will exactly cancel the positive deviations, making this method always give you zero, no matter what the dispersion is. For the slightly more advanced math to prove it:
E(X-μ)=E(X)-μ=μ-μ=0, with E(X) being the expected value of X. If you don’t understand this and want me to explain, I can.
So this isn’t gonna work. At this point you may think “OK, so why don’t we just make all the deviations positive?” In other words, instead of using X-μ, why don’t we use |X-μ|, or the absolute value of X-μ? This turns out to work, and sometimes people do use it, but absolute values are really annoying mathematically. First, absolute value doesn’t have a simple arithmetic formula, so its always kinda clunky to express. Second, and perhaps more importantly, you can’t take the derivative or the integral of an absolute value function, which are two processes that are really useful in analyzing data. These two things together make the absolute value trick less appealing. So instead, we can square the deviations, which solves all of our problems (except one, which will be shown).
Thus, the variance is the average of the squared deviations from the mean. In other words, it is the mean of (X-μ)2 .
Now, if you’ve been following along, you may have noticed a problem. The units of variance are the square of the units from my original data. So if I have a set of data measuring inches, my variance, or measure of the dispersion, will awkwardly be in inches squared? This presents an obvious problem since it takes away most of the practical applications of the variance. Thus, in most of applied statistics, the solution is just to take the square root of the variance. This is the value that we call standard deviation.
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