Average vs Mean

 

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.