Estimating Areas Under the Normal Curve Using Z-Scores

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  • 0:04 Areas under the Normal Curve
  • 0:54 Using Statistical Tables
  • 2:25 Area Between Two Z-scores
  • 4:09 Area beyond a Z-score
  • 5:23 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

So now that we have a Z-score, what is it used for? Sure, it can make your life easier when describing standard deviations, but finding the area under the normal curve is where the Z-score shines.

Areas under the Normal Curve

A Z-score is a measurement of how far away a point is from the mean in terms of standard deviations. It is calculated by subtracting the value of the mean from the value of the point in question, then dividing by the standard deviation of the set. If you have the Z-score of a data point, what can you do with it?

For starters, you could use it to refer to the points of standard deviation with fewer decimal points. However, one of the greatest advantages of having the Z-score of a set of data is the ability to find out how much of the rest of the data corresponds to a particular behavior. For example, when talking about the difference in test scores, would you rather hear a teacher repeat the standard deviation to the thousandths every time, or perhaps just say 'Z-score of 1?' To do this, we have to find the area of under the normal curve between a Z-score.

Using Statistical Tables

There are two ways of doing this. If finding the integrals of curves with respect to two different limits is your thing, then go right ahead, put that calculus knowledge to use! Just have fun trying to define C. However, if you are like the rest of us and wouldn't mind a little help every once in a while, then it's to the tables we go!

Statistics may be one of the few math fields where looking at a cheat sheet is actually encouraged. However, rather than show the answers to the problems, the cheat sheet in question I am talking about are the ubiquitous tables found in every stats book, and right here. Very often, they have extremely exciting names like Table for Areas under the Standard Normal Curve. Riveting, I know.

In any event, you'll have to figure out what kind of table it is first. Some of these show the total area from zero to the Z-score in question, while others only show from the mean to the Z-score. Here's a quick check - look at the spot for 1.00. If it is 0.3413, it is from the mean. If it is 0.8413, it is from the zero. But how do you find the spot for 1.00? Look on the axes of the graphs. One side will have units and tenths, often the Y-axis. The X-axis will have hundredths. These are of the Z-score, not the standard deviation. Multiply that value by the total sample size, and you'll know how many people meet the criteria in question.

Area Between Two Z-Scores

That was quite a lot to take in, so let's slow down and go through step-by-step. We will first work out how to find the value between two Z-scores, then we'll see how to do it for all values under or above. First, let's find the value between two Z-scores. Say that you had to find all data points between the Z-scores of 2 and -2.

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