Though a useful statistical tool to simplify things and arrive at broad 'ballpark' conclusions, the normal curve, I believe, has been mistakenly used in contexts where it needs to be 'tempered' or adjusted, or where it should not be used at all. The normal distribution curve is called the bell curve because it looks like a bell (please refer to figure 1).
The 'Gaussian Probability Function' is based on the 'Normal distribution' (which presumes that the mean of a distribution is also its median and its mode)---which in plain English means that the number of values less than the mean of a given set of values (let's say test scores) is exactly equal to the number of values greater than the mean and the highest number of values flock around the mean. It is a tenet of the central limit theorem that any large distribution of variables (e.g. number of children per family, or CAT Test scores) tends to cluster around the mean value.
Applying the curve
To standardize and use the the Normal Distribution in different contexts (heights, stock market returns, test scores, IQ scores, number of stars in a constellation) it is applied to a probability density distribution ( a graph showing the frequency or probability of different values occuring). In this, we ascribe the value of zero a 50% probability, i.e. we now can have a 50% probabilty of values more than this and less than this respectively. Then values are ascribed to different points on the curve, based on the formula z=(x-m)/σ where x is the actual value, m is the mean and σ is the standard deviation. Each value on this normal distribution curve signifies a corresponding percentage.
Using this simple mathematical tool, given the mean (average) , standard deviation and the value of the variable, we can calculate the approximate frequency of its occurence. E.g. if the mean test score is 50, the standard deviation (SD) is 20, and the test score of a student is 80, z=(80-50)/20 = 1.5. Using a normal distribution table or the =NORMDIST function on an Excel, this corresponds to a frequency of 90% cumulative---which means that about 10% of students would get more than this score, and about 90% would have got less.
The assumptions, and the pitfalls
What people often forget while applying this beautiful piece of mathematics is that the real world, unfortunately, does not always have the simple purity of the bell curve. Moreover, the bell curve makes some very simplifying assumptions, which would not always hold true in real life. For example, it is presumed that 68% of the values of a distribution would occur within +/- 1 SD of the mean (please refer to figure 1) and 99.9% of values would occur within 3 SDs of the mean. Which means that we don't expect too many 'outliers', or statistical events at the far end of the curve on either side. The presence of outliers or extreme values, leads to 'fat tails', or in statistical terms, kurtosis values of greater than 3. If the Standard normal bell curve is used ot model these events, it can lead to 'model error' (figure 2 shows a 'fat tailed' curve).
Belling the Cat with the fat tail
When the standard normal curve is simplistically used in risk management, to forecast market returns and accordingly predict capital requirements (i.e. how much own capital do I need to tide over a worst case scenario, also called 'Value-at-Risk), it can lead to disaster, as risk managers have found out as a result of the recent financial crisis. The problem is not with the concept of Value-at-Risk (VaR), but with applying the bell curve to all markets and situations. Using data from bullish markets, mean and SD values were computed. These values were then juxtaposed into the normal bell curve, further aggravating the error factor in the model. The result: many banks were undercapitalized, vis-a-vis possible (but as per their models, improbable) trading losses. Lehman Brothers is a case in the point, which went under, with just USD 6 billion in losses, though it had overall assets of...
Moreover, the bell curve is a probability distribution of independent events (e.g. test scores, where there is no cheating). Real life may create correlated events (e.g. market sentiments across asset classes are often correlated; similarly cheating in an exam can cause one bright pupil to cause the marks of numerous pupils arounf him to get good marks).
Another area where the bell curve is often purportedly applied is employee performance. This application presumes that 70% of the employees will form a category, around the average (roughly +/- 1 SD if one is to apply the normal curve) and that the top 20% of the employees will perform beyond this. It imputes these values to populations of employees. However, what should be understood that in a given 'universe' of employees, there can be plenty of exceptions. Hence, more often than not, this becomes an arbitary, relative ranking---in which case, it is a method loosely based on the bell curve (or the central limit theorem, to be more precise), but by no means is it an application of the curve itself.
Falling out of the Curve
Nassim Nicholas Taleb in his book "The Black Swan" has also written about the pitfalls of using the normal curve to predict market returns, where the actual markets may have 'fatter tails' or more returns that may be considered 'extreme' . Chris Anderson, in his book "The Long Tail" wrote about how internet marketing companies create demand for 'out of the ordinary' products---e.g. retro music. Ian Bremmer and Preston Keat have recently come up with a book called "Fat Tail - The Power of Political Knowledge for Strategic Investing" which refers to the frequency of political instability affecting investment envornments and our tendency to think that they are rare events, whyen actually they may not be that rare.
So though the bell curve is a useful tool, it must be carefully 'stress tested' to verify if the variable being 'fitted' to the curve indeed matches it. The variable must also be tested for 'trending' or external correlations which may affect it, before proper models can be drawn up. Otherwise, it may actually be better to simply chart out historical data and draw inferences on that basis, presuming everething else remains the same (as with most other things, the Romans had a pithy latin phrase for this too--- ceteris paribus).
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