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Consider the population of a typical town in U.S. The average height of an adult male is 70 inches with a standard deviation 6 inches and average height of an adult female is 65 inches with a standard deviation of 3.5 inches.

Average is the sum of height’s of all the male or female population in the town divided by the number of male or female residents in the town. Standard Deviation is a measure of dispersion, i.e. a measure of distribution of sample heights away from the mean.

   According to the theory of normal distribution, popular for it’s bell shaped curve,

68% of the population will be 1 standard deviation away from the mean (i.e. 68% of the male population in the town will have a height in the range of 69 to 71 inches ) .

95% of the population will be 2 standard deviation away from the population mean (i.e. 95% of the male population in the town will be in the range of 68 to 72 inches ).

Finally 99.7% of the population will be three standard deviation away from the population mean (i.e. 99.7% of the male population in the town will be in the range of 67 to 73 inches )

Normal distribution is useful in modeling the physical world. for example,

1. The height distribution in a given population.

2. The weight distribution of all living mammals

3. The Average of life expectancy of a population

4.  The fuel efficiency ( i.e. miles per gallon ) of compact cars

Most of the physical world follow normal distribution. Hence the bell curve has gained popularity as a reliable tool to predict and model the behavior or characteristics of a sample set or a given population. Using Bell Curve (i.e. Normal Distribution ) we can provide answers to problems similar to the one below.

An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will burn-out within 365 days?

Answer : Using Normal Distribution we can predict that there is 90% chance of the bulb burning out within 365 days. Your can find the complete solution here.

Now let’s consider the frequency of occurrence of outsized events in the stock markets. In a world described by the bell curve (also called “mild randomness”), most values are clustered around the middle. 95.4% are within 2 sigmas from the mean. The average value is also the most common value. Outliers contribute very little statistically

In the last two decades the U.S stock market has experienced what would have been considered a 3 sigma or 4 sigma event based on Normal Distribution. According to the chart above, these 3 sigma or 4 sigma events have a probability of occurring once every 740 years and 31,575 years respectively. This proves that the tail of the bell curve can be significantly fat.

To conclude :

Bell curves are ideal to predict or model normal events, but there are events that are beyond the realm of normal distribution. These events  defy the laws of physics. They are captured by power law.

As Prof Sanjsy Bakshi suggests.

In a universe described by the bell curve, experiencing “mild randomness”, one single observation, such as a very tall person, may seem impressive by itself but will not  disproportionately impact the aggregate or the average.Such randomness that disappears under averaging is trivial and harmless because you can diversify it away.

In a universe dominated by power laws, where black swans proliferate, outliers are more common than bell curves can predict. In this exotic world winners tend to take all and the rest get nothing.

Many investors lost a lot of money in the 2008 stock market crash when they relied on normal distribution to model and predict stock markets which follow the power law.

So the big lesson is that you have to acknowledge the presence of black swans – both positive and negative, and organize your life accordingly. You cannot rely on models based on the wrong worldview of a bell curve universe.