In the financial markets, the definition of volatility is assumed to be standard deviation. However, in practice, people seem to either forget the definition or completely substitute it with the mean absolute deviation.

For example, a time series of 29 zeroes followed by a single value of 1000 (i.e., 0, 0 ,0... , 1000) has a standard deviation that is 5x greater than its mean absolute deviation.

Of course this example is an extreme, a fat tail distribution, but even in normal bell-shaped distributions, standard deviation is about 20% greater than mean absolute deviation.

we don't know sigma

To test the hypothesis that we are all confused about this, Daniel G. Goldstein and Nassim N. Taleb asked a simple question to 97 portfolio managers, 13 Ivy League graduate students preparing for a career in financial engineering, and 16 investment professionals working for a major bank. Some folks in the first group did not turn in their answers so, in total, there were 87 respondents.

The question provided average returns and average move data, and asked for daily and yearly standard deviations.

Of the 87, only 3 got to the correct daily standard deviation, and none provided the correct yearly standard deviation. The majority responded with the average move (i.e., mean absolute deviation).

but we know the formula!

After the study, when respondents were presented with their error, they rarely had an understanding of it. However, when asked to present the formula for standard deviation, they expressed it correctly.

Disconcerting but not surprising. Reminds me of Richard Feynman teaching a class at MIT and, for fun, telling his students how their French curve, at the lowest point, no matter how they turned it, would have a horizontal tangent.

... and I picked up my French curve and began to turn it slowly. “The French curve is made so that at the lowest point on each curve, no matter how you turn it, the tangent is horizontal.”
All the guys in the class were holding their French curve up at different angles, holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. They were all excited by this “discovery” -even though they had already gone through a certain amount of calculus and had already “learned” that the derivative (tangent) of the minimum (lowest point) of ANY curve is zero (horizontal). They didn’t put two and two together. They didn’t even know what they “knew”.
Surely You're Joking, Mr. Feynman!

references

Goldstein, D. G. & Taleb, N. N. (2007). We don't quite know what we are talking about when we talk about volatility. Journal of Portfolio Management, 33(4), 84-86. [Download]