## Volatility of Rank

To understand this calculation, it is necessary to recall that there are two ways of measuring the "instability" of a sequence of numbers:

• Variance and Standard Deviation

Here, one takes the mean of the numbers, then takes the square of the distance of each individual number to that mean, then looks at the mean of those squared distances. This gives the variance. The standard deviation is simply the square root of the variance. • Volatility

Here, one takes the median of the numbers, then takes the absolute value of the distance of each individual number to that median, then looks at the median of those distances. This is the volatility. Thus, the two measures differ only insofar as variance uses the mean and the squared distances from the mean, whereas volatility uses the median and the absolute value of the distance from the median.

In the case of the manager ranks in our manager vs. universe calculation, we use the volatility rather than the variance. This is because we are already displaying the median of rank (because we think that this is the appropriate measure of "average" in this case). Obviously, the volatility is a better companion to the median than the standard deviation would be.

Volatility of rank = median(i = 1, ..., m)( |Ri - median(i = 1, ..., m)( Ri)| )
where
m = number of dates
Ri = manager's rank number for the i-th date

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