How do you determine whether you can use the central limit theorem in a particular problem?

Answer 1

I don't have a simple answer. I will give examples where the central limit theory seems to fail. From Wikipedia (http://en.wikipedia.org/wiki/Central_limit_theorem) From another viewpoint, the central limit theorem explains the common appearance of the 'Bell Curve' in density estimates applied to real world data. In cases like electronic noise, examination grades, and so on, we can often regard a single measured value as the weighted average of a large number of small effects. Using generalisations of the central limit theorem, we can then see that this would often (though not always) produce a final distribution that is approximately normal. Let me restate the idea of the central limit theorem: When many small, independent and random outcomes are summed, the result will eventually be normally distributed (bell shaped). The underlying processes which produce the outcome must be stationary (not changing). We state that the mean of a sample should have a normal (bell shape) distribution, if it came from a random sample. Again, the underlying population must be stationary (unchanging properties). 1) The Stock Market is an excellent example of where the central limit theory does not apply, due to the problem of non-stationary and dependent outcomes. A stock with 100 year price history does not permit me to predict the future price with a normal distribution. 2) Public opinion polls regarding politics frequently do not adhere to the central limit theory, because people are continually reacting to the media. A larger sample, taken over months, may be less reliable because people change their mind. 3) Many human traits are not the result of small random and independent factors, but of many factors interacting with each other, thus do not adhere to the bell shape curve. The quantity of alcohol we consume probably does not fit well a bell shape curve, because for a certain segment, they are addicted to alcohol.