MORE ON THE DISTRIBUTION OF THE MEANS

It is understandable that certain variables , such as height and weight, have distributions that are approximately normal. We know that most of the world is pretty close to average and that the farther we move from average, the fewer people we find. But why does the distribution of sample means look like a normal distribution?

This remarkable fact is explained by the Central Limit Theorem. The Central Limit Theorem says that for samples of a sufficiently large size, the real distribution of means is almost always approximately normal. The original variable can have any kind of distribution. It does not have to be bell-shaped in the least. ("Real" distribution means the one you would get if you took an infinite number of random samples. The "real" distribution is a mathematical concept. You can get a pretty good idea of what the "real" distribution looks like by taking a lot of samples and examining plots of their values ” as we have been doing.) Sufficiently large size ? What kind of language is that for a mathematical theorem? Actually, our paraphrase of the Central Limit Theorem has several vague parts . You have to say what you are willing to consider "approximately normal" before you know what size sample is "sufficiently large." How large a sample you need depends on the way the variable is distributed. The important point is that the distribution of means gets closer and closer to normal as the sample size gets larger and larger ” regardless of what the distribution of the original variable looks like. Ultimately, the means will look like a normal distribution. That is why the normal distribution is so important in data analysis. Your variable does not have to be normally distributed. Means that you calculate from samples will be normally distributed, regardless. If the variable you are studying actually does have a normal distribution, then the distribution of means will be normal for samples of any size. The further from normal the distribution of your variable is, the larger the samples have to be for the distribution of the means to be approximately normal. This is a fundamental assumption under which Statistical Process Control charting operates.