Standard deviation

We can use the standard deviation to see how far from the mean data points are on average. A small standard deviation means that values are close to the mean, while a large standard deviation means that values are dispersed more widely. This is tied to how we would imagine the distribution curve: the smaller the standard deviation, the thinner the peak of the curve (0.5); the larger the standard deviation, the wider the peak of the curve (2):

The standard deviation is simply the square root of the variance. By performing this operation, we get a statistic in units that we can make sense of again ($ for our income example):

When we moved from variance to standard deviation, we were looking to get to units that made sense; however, if we then want to compare the level of dispersion of one dataset to another, we would need to have the same units once again. One way around this is to calculate the coefficient of variation (CV), which is unitless. The CV is the ratio of the standard deviation to the mean:

Since the CV is unitless, we can use it to compare the volatility of different assets. So far, other than the range, we have discussed mean-based measures of dispersion; next, we will look at how we can describe the spread with the median as our measure of central tendency. Our next post will focus on Interquartile range and related topics