Is Being Average "Normal"?

     Many people are fascinated by surveys and tables of statistics. They can be fun -- and even useful, when interpreted by people who know what the data does, and does not, indicate. They can also be misused by presenting data in such a form that people will jump to conclusions that are not really supported by the data. This is done directly by many politicians and for political purposes and causes.
     I have mentioned before the quote popularized by Mark Twain -- "There are three kinds of lies: lies, damned lies, and statistics." I like statistics because I am fascinated by numbers. Their relationships and the ability to use various formula upon the data is a lot of fun. But I don't have sufficient training to use (or misuse) them properly.
     But this particular blog is not about such weighty matters. It is more about how people can take "everyday" data and make more of it than what they deserve. In particular, it is about averages and the definition of normal. An average is a fact when you are dealing with numbers. The average of two, five, and eleven is six.  The average height of a human male in the United States is approximately 69.7 inches (177 cm) -- but there are not that many men in the United States who are exactly 69.7 inches tall. In other words, few men are of "average" height.
     Averages only make sense for characteristics when you examine them within a distribution. A normal distribution curve looks like:

     Normal means, basically, that the curve has a uniform change throughout the graph -- there are just as many above the average as there are below the average and the percentages change consistently on both sides of the "bell curve". This ideal curve doesn't occur that often in nature. One problem that happens is that there are often assumptions that a distribution of data will meet a normal distribution without doing a sufficiently wide sampling of data to justify it. For example, the data for men and women's heights in the U.S. can look like:

     Note that the peak for men ends up around 70 inches, while the peak for women is at about 64.5 inches (164 cm). The peak is higher than for men and the curve is narrower. This indicates that women's heights don't vary as much as men and concentrate at the average. However, in both cases the curve is very "smooth" and symmetrical. For height, it seems that the data really does seem to support this -- with a longer "tail" at the tall end indicating a slightly greater number of very tall people versus the number of very short people.

     Any characteristic that can be measured can have an average. Skin color is based on the amount of pigmentation from different types of melanin in the skin. There could be an average value for this. The average weight for a U.S. male (in 2015) was 195.5 pounds (88.5 kg) -- an increase of 30 pounds (13.5 kg) since 1960. The number of hair follicles per square inch can be measured -- thus, there is an average value for "hairiness". Ear size can be measured from top to bottom (or front to back or distance out from the skull).

     How many people have an average height, weight, skin color, hairiness, and ear size? Probably only a half dozen or so within the United States. Are these six people the "normal" ones? No, not even from a statistical sense because there are going to be many other measurements that are potentially able to be done -- eye color, IQ, foot size, hand span, distance from bottom of nose to top of lip, and so forth. Everything that can be measured can have an average -- but none of them are "normal" because humans are not just one characteristic. They are combinations of many, many characteristics and there may be some that are genetically linked (if you have one then you also have the other) but most appear to be totally independent. Skin color is totally independent of IQ. Eye color appears to have little to do with height. Weight and height do have some correspondence but it is possible to have a tall thin person as well as a short heavy one.

     Since being average in every possible measurable area is highly unlikely,  it is certainly not normal to be average.