Re: Randomness give a bell-curve. If then something is following the bell-curve

Hi!

If randomness is ruling a variable it would follow the bell-curve. I`m aware of that logic donīt allow to invert the the "argument", but couldnīt you to some degree take it as a reasonable sign that if a variable follow the bell-curve it is because of randomness in the measured variable?

Sincerely Mikael!

One of the most important theorems in probability theory is the central limit theorem. Roughly speaking, this theorem states that if you add up a bunch of indentically distributed random variables then the resulting sum will have a distribution similar to a normal distribution. A side note: mathematictians tend to use the term "normal" distribution while the general public (and some social scientists) refer to it as a bell-shaped curve because if you plot the distribution it looks somewhat like a bell (see here for a picture). Because of its importance, this distribution gets a third name from physical scientists, who tend to call it a Gaussian distribution.

In your question, you are asking the opposite of the central limit theorem: given data (such as heights of adult men age 25 in India) that looks normal (that is, has roughly the same distribution as the normal distribution) when plotted in a histogram, can we say that there is some random source producing the effect? In logic, this is an example of the converse. If A implies B, then the converse statement is B implies A. But do we have any reason to believe that the converse should be true?

As you point out in your question, mathematically, the converse is not always true. All Russians are people, but it is not true that all people are Russians. However, in the scientific method, often the converse of mathematical statements are used as supporting evidence for a theory. For instance, it can be shown mathematically that if gravity obeys an inverse square law (and Newton's laws of motion are true), that the planets move in elliptical orbits around the Sun. Now in point of fact, the planets can be observed to move in elliptical orbits around the Sun. The converse statement that planets moving in elliptical orbits around the Sun implies that gravity obeys an inverse square law may not follow automatically from a mathematical standpoint, but it seems a reasonable step to take when studying the motion of the planets. We say that the inverse square law together with Newton's laws of motion explains the motion of the planets are some level.

In the same way, we can say that the central limit theorem gives (at least a partial explanation) for why we find normal distributions so often in nature. Often, randomness just means uncertainities about the history of the data that might affect the outcome. For instance, the height of the parents of an individual in question, the individual's diet, etc., all contribute to the final height. In a world of imperfect information, these are unknown, we might label these effects as "random". And the central limit theorem tells us that the sum of random effects will tend to produce normals. Without even saying what the random effects are exactly, the central limit theorem acts as an explanitory tool for studying patterns that we find in data.

Your question really hits the heart of the philosophical question of what it means for an explanation to be a scientific explanation. Pages like this one or this one give more thoughts (and many links) on what constitutes scientific explanation, but of course they only scratch the surface of the work that has been done in this area.

Mark Huber