What is Correlation?

Correlation; does “A” mean a certain level of “B”, or vice versa? Regression analyses are used to find relationships between two variables. The trick to a healthy analysis is taking a logical standpoint with the data; the phrase “correlation is not causation” comes to mind. Two variables can be seemingly perfectly related because of a regression analysis, but have absolutely no connection whatsoever. In some way, this was the error found in Darwin’s published theory of evolution. The “Rule of Three” is a variation of the misconception that relationships are in fact related. This is a brief understanding of what regression specifically covers in the field of statistics; there is much more behind it though.

A commonality has revealed itself among every statistical theorem, calculation, or concept: the universe corrects itself. That sounds like a lofty idea, almost philosophical or religious. Mathematically you can understand it as the equilibrium or statistical mean. The normal distribution shows that statistically, populations will fall under the same distribution curve, with some minor variations. The law of large numbers states that given enough trials, occurrences with equal probabilities will even out. The Central Limit theorem says that with enough samples, the distribution will be essentially the same as the population; which, if large enough itself, would be close to a normal distribution. And finally, regression. Francis Galton initially developed the mathematical process of regression analysis to show that sons’ heights regressed towards the mean height of men rather than towards a more hereditary direction. 

There is an interesting way in which another concept applies the of coming back to the center point, or a normal distribution; the solar day. This is the definition of a day that we currently use for having a standard, non-variant unit of measurement. In reality, every day is different, the time it takes the earth to rotate around the sun varies slightly, and due to the angle at which the earth is tilted relative to the sun, the position changes as well. This forms a solograph, which looks quite similar to a statistical image we should be awfully familiar with now. The important aspect of this anecdote is not however that the image looks like a normal distribution, what I am more interested in is the suns position during one specific time each day. 

This image is created by taking a time-lapse photo of the sun’s movement across a fixed point. While the sun may always reach a peak point, it will not always be at the same time each day. Over the course of a year, peak sun-time will appear to be faster or slower; in this case, the sun is not moving variable, it is simply the crux at which we base earth’s movement off of.

The last idea that brings home the idea of regression towards mean, is the use of statistics, albeit unknown at the time, in the definitions of recorded time, calendars; specifically, the creation and changes of calendars throughout history. Before there was a great deal of communication across cultures, everyone had a different system of time demarcation. It wasn’t until the use of sundials was widely deemed unnecessary that cultures merged and adopted standard measurements. The reason behind needing a calendar, aside from knowing how soon Christmas is, was to form a standard, unchanging measurement of time, in terms of days, months, and years. That is exactly what we have now, sort of. We are always trying to make the year fit as close to the norm as possible, which is why leap days came into effect. Every four years, March 1st occurs one day too soon. To fix that, an extra day is added to the end of February. That solves the problem, sort of. It actually overcorrects slightly. In the Gregorian calendar, which we use today, 3 leap days are left out every 400 years. 

Statistics does not create information, it reformats it, makes it clearer to understand. It is simply another tool used by humans to find meaning in the great complexity of the universe. If I have learned anything from this, it’s that the normal distribution is the answer to everything ever. There is always balance in the force.

Reference

Blanco-Muriel, M. et al., (2001) Computing the Solar Vector. Solar Energy. vol.70, i.5, pp. 431-441