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Question 779:



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  1. Q779_1_q779_shapeofZs.xls
Let me rephrase the question a bit. It sounds like you're asking if you take any raw set of data, then convert the raw scores to z-scores, the shape of the distribution of z-scores will be bell-shaped. The answer is no, the distribution might change, but it doesn't mean it will always be bell-shaped. We can quickly prove this. Let's take a set of 30 numbers and convert them to z-scores by subtracting off the mean and dividing by the standard deviation.  See the attached excel file for the numbers and the resulting 2 graphs. What we see is that the distribution of the z-scores is a bit different, but far from bell-shaped.

When we convert data using z-score, we're turning the raw data into a standard normal distribution. A standard normal distribution is just a normal distribution with a mean of 0 and standard deviation of 1. The z-score is just a linear conversion. If your raw data is not normally distributed, then your z-score distribution will also NOT be normally distributed.

A related concept to this which you might be thinking of is called the central limit theorem. In short it states that if you were to take a bunch of sample means from a population, the distribution of the means would be normally distributed (bell-shaped) regardless of the parent population shape. So if you have a skewed population or a population with a binomial distribution (only 2 outcomes), the shape of the sample means are normal. That's a fundamental topic in statistics.

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