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Fundamentals of Statistics 2: The Normal Distribution :: Why do you need a table to find probabilities with z-scores?
You might be wondering why it's necessary to convert your normal data into a standard normal data then go through the trouble of looking up values in a table or calculator?  One questions I get a lot is : What's the formula for finding the probabilities for a z-score?

Photo By: foundphotoslj
Well, if this were the normal rectangle we could just use the length and width to get the area; but this is a curved bell shape so we have to use calculus to integrate the area between two points under the curve to approximate the area instead. Doesn't that sound like fun?  So there is no simple formula like length time width or a2 + b2 = c2 since the area under the bell curve will be different depending on where you pick (unlike a rectangle which would have fixed area between any points).

So it turns out that getting the area mathematically is a rather complicated procedure. Even if you've had some calculus you can't use the normal integration techniques. Online calculators like the one I created use an approximation to find the area accurate to several decimal places. Before the days of high-speed computers, really smart mathematicians did the calculus and tabled the values using the standard normal form we discussed.  That's why you need to lookup values in normal tables.

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