Out of curiousity, I am taking the 5 minute movements of various stocks (like Delta) and calculating their percent change and then taking all of those changes and graphing them in a histogram with categories like number of movements between -95% -100%, -90% to -95%, etc ... Now I've got a really nice bell curve going on, but with an interesting difference. The tails at the end of the curve are higher than they should be. Any ideas why?

Financial instrument return distributions are subject to more extreme events than is the case in a "standard normal distribution"... this statistical phenomenon is called leptokurtosis and more generally referred to as "fat tails"...

correct; the normal distribution is the reason why LTCM went down, they forgot about the black swans.

It is interesting that aphexcoil posed this question after observing five minute charts. I wonder if fat tail are observable on all time frames, then I wonder if that means that the intraday strategy I employ is correct. How would aphies' studying look compared to say just putting up bollinger bands on the stock. Would you set say 2std setting on a five minute chart going back the same look back period as aphie? Would you see a few too many piercings of the bands and consequently conclude fat tails exist? I mean if you could use the standard 20 period setting, it would seem to me that there are super fat tails on some stocks. Which brings to the point what are the correct settings to make academically correct observations? How about profitable observations. I realize these are the answers that can break a hedge fund so please do not get in a tizzy, I was just throwing those questions out there.

In one of the market wizards book (The New Market Wizards, I think) Meister Eckhardt argued that statistical measures could not be used with the stock market, atleast in a classical sense. I am pretty sure he even mentioned the bell curve specifically as being useless. I am not an expert here, but what I got was that a bell curve assumes a distribution which cannot be assumed in the stock market. I spoke with a friend about this a few months ago who has had more experience with statistics than I. From our conversations I gathered that many statistical tools assume some basic distribution, or a fixed set of probabilities. For example, one can flip a coin 100 times and measure the deviation in percentage from 50/50. The sample (100 in this case) will more or less conform to 50/50. The degree to which it can vary is unlimited (possible to get 100 heads) but the greater degree of variation, the more unlikely it becomes. How do we know what the "normal" distribution of market prices is? The likelihood of Microsoft trading at 100 or 20 seems to have no fundamental basis. Statistics as applied to this field, I think, must either be applied using different assumptions or not applied at all.

The bell Curve is as useless as the straight line in gravitation's law so ignoring the bell curve would like ignoring the Newton's First Law of Motion which is the law of inertia: "An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force." I don't think that physicians consider this law of inertia as useless although it is the simplest so I don't think that the bell Curve should be considered as useless it is rather that people misused it. It is not a law to use directly it is a law of reference. Unerstanding the basic law is essential. See for example the post on Deming's SOPK <a href="http://www.elitetrader.com/vb/showthread.php?s=&threadid=16751" target="_blank">"Deming's SOPK: System of Profound Knowledge "</a> or directly: http://www.maaw.info/ArtSumDeming93.htm (what Deming calls "special cause variations" is the analogy of unbalanced force above): "Chapter 10: Some Lessons in Variation The purpose of this chapter is to provide: 1) some easy lessons in variation including examples of situations where common cause variations are confused with special cause variations, and 2) some illustrations based on the concept of a loss function. Deming explains that variation is life. Life is variation, but those who have no knowledge of statistical theory tend to attribute every event to a special cause. One qualification useful to anyone, and definitely needed by anyone in management, is to understand the concept of variation. This understanding of variation will help them understand the system and to stop asking people to explain the day to day, month to month, and year to year ups and downs that come from the variation that is built into the system."

I should have known your inability to write would preclude an inability to read. For those interested, neither Meister Eckhardt nor myself proclaimed the Bell Curve useless in general. Rather, the usefulness of the bell curve rests on what one can derive from its application. One basic tenant of bell curve theory is that a proper sample will result in a distribution of occurrences about a mean. If there is no mean, then the bell curve is useless. What I think Eckhardt was stating was that the stock market has no mean. Or, that the historical mean of stock market prices has little to no bearing on the future mean. An example: Let's say the mean historical price of IBM = $30. Does one then assume because IBM is trading at $50 that it MUST fall back to $30? If it is trading at $10, must it rise back to $30? What if it goes to $0? If the company no longer exists then using a mean for predictive purposes or otherwise is worthless. The example I had employed of flipping a coin was one with an invariable outcome. If one were to flip a coin an infinte number of times the outcome would be 50% heads and 50% tails. I do not think such an invariable outcome exists with the markets, and thus using a mean, or the bell curve, for predicting changes in the price of stock would be useless.