Average Guy
I ain't no Christian or no
born again saint
I ain't no cowboy or Marxist D.A.
I ain't no criminal or Reverend Cripple from the right
I am just your average guy, trying to do what's right
What is the average?
Humans have an innate ability to solve complex mathematics. That however is not exceptional. The spiral of sunflower seeds follows a Fibonacci sequence. A lifeless coastline follows the fractal pattern
of a Mandelbrot set. If plants and lifeless
coastlines follow mathematical rules, then maybe humans are not so exceptional. It may be only the language and symbols of mathematics
that are intimating and confusing.
Mathematics uses
words that may seem familiar but may have more nuances than the conventional usage. In mathematics, average means the centrality of the normal.
Humor is often
found in the conflict between what is said and what is meant. Two jokes illustrate the difference between
average in common usage and in mathematics.
A once popular
radio show popularized the Lake Wobegon
effect, where “all the children are above average”. The joke being that the average is the centrality
of the normal. If everyone is above average,
then that is no longer the centrality and it is time to compute a new average.
Another joke
is that a statistician who has his head
in an oven and feet in ice is supposed to say that on average his body temperature
is at room temperature, and that is “normal”.
The problem is that what is considered normal to most people, is a Gaussian
distribution to a statistician. This distribution
has a centrality of zero AND a variance of 1. Room temperature is a normal distribution.
The variance of having your feet in ice
and head in an oven is much greater than 1, the variance of a normal
distribution and no statistician would make the statement in the joke.
When people
say average, it is often assumed that they intend the mean. The mean is easier to compute. It requires the ratio of only two numbers,
the total of the observations and the number of the observations. The median is harder to compute. It requires identifying the point at which
50% of the observations are above, and 50% of the observations are below. You
can calculate the mean from the totals.
You need to sort each observation to compute the median.
A perfect uniform
normal distribution has a mean, median, and mode of zero.
That is not terribly useful, but if the mean is added to every observation,
the coordinate system is translated to a new origin, which is the mean
and not zero. If the median still
equals this mean AND the variance is 1, then, and only then, this is a coordinate translation
of the “normal” distribution. If the mean
and median are not equal, then the observations are NOT “normal”.
A Gaussian distribution is commonly called “normal” because
nature appears to favor this distribution .
People judge themselves against the average, the centrality of the normal.
A problem may be that the “normal” is confused with the mean, because the median may be the harder to compute. The mean and median income, or wealth, in the United
States are very different. A statistician
would say that is not “normal”. Nature, or humans, might not do all of the
computation that a mathematician does, but you only to do those computations to
prove that it is normal. Humans, and nature, can apparently do those computations
innately to tell if it is normal.
Average is the centrality of a normal distribution. If the mean and median are very different from
each other, then the distribution is not normal. When the distribution is not normal,
the average might be the median, and not the mean.