
Now remember, when we calculate the z score of any value in the distribution,

we first subtract the mean which shifts the distribution without changing the

shape so that zero is now the mean. And when we divide by the standard

deviation, we then change the shape. Let's look at it this way. We have any

distribution, with mean, mu, and standard deviation, sigma. Which basically

means that sigma is one standard deviation away from the mean. After we

standardize this distribution, what is going to be the zscore of sigma? Well,

remember when we subtract mu, we shift it so that mu is now zero. So now, the

zscore of sigma is going to be sigma minus zero divided by sigma, which is

sigma divided by sigma, which is 1. So, the zscore of any value, that's one

standard deviation away from the mean, will then be 1 after we standardize it.

Which means that the new standard deviation of this normalized distribution, or

standard distribution, is 1.