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Fundamentals of Statistics 2: The Normal Distribution :: The Standard Normal Distribution and Z-scores
If we want to know the area under the normal curve with more precision than 1,2 or 3 standard deviations we can use the z-score to look-up the area in a table. Recall that the normal distribution is a family of distributions (quite a large family in fact since there are an infinite number--each defined by their mean and standard deviation) so how can we tell if our distribution will be like a tabled value?  To solve this problem we just convert our data to a standard form.

The standard normal distribution is just a normal distribution with a mean of 0 and a standard deviation of 1.  There's nothing terribly special about this standard form. It's just simpler and easy to remember. To convert our normal distribution to the standard normal form we use the z-score formula :

All that's going on in the formulas is we're dividing the difference from a data-point to the mean by the standard deviation. That means we're saying how many standard deviations a value is from the mean. If you're an adult male in North America and are 72 inches (6'0) we can find out how many standard deviations you are above the mean by using the z-score formula: 

z = (72-70)/4 = .5

At 6'0 you would be half a standard deviation above the mean. Now we need to know the area that corresponds to a z-score of .5 or .5 standard deviations above the mean.

By the way, this same approach (knowing how many standard deviations a point is from the mean) will be used when we discuss the central limit theorem and the 1-sample z test. It helps explain why we're able to know so much about a population just from a sample. So pay attention, here (stats professors love to ask about z-scores!)

I've put together an excel calculator which will do almost anything you want to do with z-scores. You can watch a short-overview of that below.

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