In the belly of the beautiful bell curve

An average of a set of observations is an alluring metric and many decisions are currently made on this wonderfully and holistically sounding statistic. Hypothesis testing is often conducted based on the assumption of normality when observations are presumed to fall on both sides of the average in a beautiful bell curve. Many scientific disciplines – medicine, engineering and economics – routinely employ averages and normality assumptions in making critical decisions.

However, averages can be quite deceiving when the normality assumption does not hold. It has been said that the average IQ in the living room with few people dramatically shot up whenever Einstein walked in. Similarly, performance of many funds is tolerable if you remove a few extreme observations. Weather is forecastable, except when it is not. Diseases can be conquered with medicines that sometimes kill you. Similarly, if earth’s polarity shifted many times in less than 10K year cycles and there were few episodes when nothing happened for millions of years, average time between episodes will be high but that is not comforting.

In a histogram of historical observations, it is often the case that the average is significantly shifted to the right of the mode (most likely outcome). Both the use of averages in decision-making as well as the use of hypothesis testing based on normality assumptions is fundamentally faulty as most events show skewness in the time between occurrences as well as the severity of an occurrence. In business, financiers and economists often use average cash flows in calculating a net present value of an investment or stock. In pharmaceutical research, bell curves are used to accept outcomes by discarding the probability of extreme events as small. In engineering, failure data provides a safety margin based on averages. In all these cases, the decision-maker is being coxed into a level of comfort by the misguided power of normality statistics. None of these decisions are likely optimal as they are based on the wrong assumptions. Higher moments (skewness and kurtosis) are necessary ingredients in making any decisions based on historical observations.