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I am studying the statistical distribution of the size of crystals formed in a cement clinker. I expected a normal Gaussian curve but it was more like a Y=1/X curve. Why is it so?
Question Date: 2009-04-14
Answer 1:

Gaussian distribution's occur when there is an expected value for the "random" object we're measuring. In this case, you're measuring the size of the crystals. It's true that a completely random number would have a gaussian distribution centered about the mean. However, the size of the crystals is not completely random!

The crystal size depends on the properties of the atoms that make up the crystal. It's possible that the atoms could arrange in different patterns. Also, different atoms and molecules will have stronger and weaker bonds. Therefore, some materials will make crystals better (or worse) crystals than other materials. This can have a large affect on the distribution of sizes of the crystals.

The most likely thing that would happen is that the crystals would start to break or fall apart once they get too big. This would have the effect of causing the distribution to get "crunched up" closer to the mean, due to the breaking of the larger crystals. This would have the affect of making the distribution look more like a 1/X curve, as opposed to a gaussian distribution.

Answer 2:

The way that crystals form is that molecules attach to a substrate, which can be another crystal or just the edge of a container. If a molecule attaches to a crystal, the crystal will get bigger, but if it attaches to the substrate, it will make a new crystal. As a consequence, a random distribution of crystal sizes is formed, just as a consequence of the probability distribution that the chances are always that a molecule will land on the wall of the container and not on another crystal.

You could conduct a very similar experiment by dropping marbles at random into a large number of different containers. If the number of containers is as large or larger than the number of marbles, you will duplicate the same distribution.

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