# Why we sometimes use mean and sometimes use standard deviation to measure volatility

Both the metrics `average true range` and `historical volatility` measure volatility of a financial instrument's price. Average true range measures volatility on an intraday basis, while historical volatity measures volatility on an interday basis.

Why then for average true range do we compute a mean, whereas for historical volatility we compute a standard deviation?

• `historical volatility, we use standard deviation` Volatility is always standard deviation. Where did you get this information from, very curious ? Feb 16, 2015 at 9:20

I just went and learned the definitions of these terms, and it appears that the difference is not that one does not use a mean but rather that they apply a different number of means.

• either statistic can be used for both intra- and inter-day measurements

• `historical volatility` is a deviation measurement (such as standard deviation) that applies two separate mean computations: the first determines the mean of all the values, and the second determines the mean of the individual differences from the first mean.

• `average true range` is a deviation measurement that applies only one mean computation: a bunch of mean-less deviations are computed and then their overall mean is determined.

So both involve taking means (because that's the only way to compute a statistic over a possibly variable number of data points), but historical volatility also uses a mean to compute the individual data points in the first place.

(Fascinating side note about the standard deviation is that you can compute it without computing the overall mean at the beginning, so technically it also involves only a single mean.)