1
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Now remember, when we calculate the z score of any value in the distribution,
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we first subtract the mean which shifts the distribution without changing the
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shape so that zero is now the mean. And when we divide by the standard
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deviation, we then change the shape. Let's look at it this way. We have any
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distribution, with mean, mu, and standard deviation, sigma. Which basically
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means that sigma is one standard deviation away from the mean. After we
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standardize this distribution, what is going to be the z-score of sigma? Well,
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remember when we subtract mu, we shift it so that mu is now zero. So now, the
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z-score of sigma is going to be sigma minus zero divided by sigma, which is
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sigma divided by sigma, which is 1. So, the z-score of any value, that's one
11
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standard deviation away from the mean, will then be 1 after we standardize it.
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Which means that the new standard deviation of this normalized distribution, or
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standard distribution, is 1.