Normal III

 

Nothin’ From Nothin’ Leaves Nothin’

Nothin' from nothin' leaves nothin'
You gotta have something' if you wanna be with me
Nothin' from nothin' leaves nothin'
You gotta have something' if you wanna be with me

What is normal?

A statistically normal distribution is one where the Median is equal to the Mean. However a normal distribution also requires that values less than zero, nothing, be allowed. In the real world less than nothing is not allowed. I mean an absolute. For example, a temperature can be below zero on the Fahrenheit scale, but it can not be below absolute zero. Therefore normal-like distributions such as the exponentially modified Gaussian (normal) distribution are used. In that distribution the Mean is close to, but not equal to, the Median. It also allows for a non-zero Skew which is not allowed in truly normal distributions. However there are still rules. An exponentially modified Gaussian distribution has a limit on the Skew of 0.31.

Another feature of a normal distribution is the 68/95/99.7 rule. That is one Standard Deviation from the Mean has 68% of the values; two Standard Deviations from the Mean have 95% of the values; and three Standard Deviations from the Mean have 99.7% of the values. If we assume that 99.7 % is close enough to 100%, it can be said that any modified normal distribution must have a Standard Deviation that is the Mean divided by three. If this is the Standard Deviation, then it is possible to estimate the Skew using Pearson’s Second Coefficient of the skew, and the Mean can not exceed 1.5 times the Median. If the Mean is more than 1.5 multiplied by the Median, then the group has to be considered as having more than one normal subgroup.

For example, LIS, the Luxembourg Income Study,  reports Mean and Median income. According to that data, the Mean US income in 2020 was $53,580 and the Median US income was $42,763 and unlike most counties the gap was increasing.  According to the Credit Suisse Wealth Databook, the US  Mean wealth was $ 505,421 and the Median wealth was $79,274.  These might be a lot of things, but they aren't normal or trending to normal.  However the mean life expectancy in the US in 2019 was 76.3 years while the median life expectancy was 78.8 years.  Life expectancy is probably one normal group. It is said that there are only two things that are certain in life: death and taxes.  Death may also be normal, but taxes may not be normal.