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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

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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.

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 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.

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 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.

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.

- Go to your chart. Look at the value for 2.
- Figure out what kind of chart you're working with and make the appropriate adjustments.

It is important to understand what kind of chart you are working with. If it measures from the mean, that's all you need for now. If it is one that works from zero, you need to subtract 0.500 from that value. This makes up for everything on the other side of the mean. In our example, 2 and -2 have the same value: 0.4772. Double it, since you are working on both sides of the mean, and you'll end up with 0.9544, or approximately 95%. - Multiply the percent that you found by the sample size, and you'll have the area.

But what if you were trying to figure out from between Z-scores of 1 and 2? Well, you'd go back to the chart. Find the Z-score for each. In this case, it doesn't matter which chart you use, but it's best to get in the habit of checking regardless and subtracting 0.500 if necessary so as not to mess up in other situations. Subtract the value for the smaller Z-score from the larger Z-score - meaning 0.4772 minus 0.3414. That means around 14%, which is the correct answer.

What about situations that involve calculating the total area under or above a Z-score? In that case, we have to flip our chart rule. If the area includes the limit back to zero, we just use that value. Otherwise, we have to add 0.500 to make up for the fact that we are going all the way to zero. So, let's say that you wanted to know the total number of people in a population of 500 who had a height of over 72 inches. The mean height is 66 inches, and the standard deviation is four inches. First, find the Z-score, which is 6 / 4, or 1.5. Go to your chart. Because this is an over/under problem, remember that it is the total back to zero. For a Z-score of 1.5, the proportion of area is 0.9332. However, that is everyone under the height of 72 inches. To get the total area at the end of the curve, subtract 0.9332 from 1, giving us 0.0668. Multiply that by the sample size of 500 and you get 33 people over 72 inches in height.

In this lesson, we learned how to use the **Z-score** to find the area under the curve. Remember that the Z-score is just the term given for how many standard deviations something is from the mean. The mean is the highest point of the normal curve, and higher Z-scores represent less and less data. By using charts or calculus, we can find the percentage of area under the normal curve. We can then multiply that value by the total sample size to figure out how many people meet certain criteria.

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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

- Go to Probability

- Graphing Probability Distributions Associated with Random Variables 6:33
- Finding & Interpreting the Expected Value of a Continuous Random Variable 5:29
- Developing Continuous Probability Distributions Theoretically & Finding Expected Values 6:12
- Probabilities as Areas of Geometric Regions: Definition & Examples 7:06
- Normal Distribution: Definition, Properties, Characteristics & Example 11:40
- Finding Z-Scores: Definition & Examples 6:30
- Estimating Areas Under the Normal Curve Using Z-Scores 5:54
- Using the Normal Distribution: Practice Problems 10:32
- Using Normal Distribution to Approximate Binomial Probabilities 6:34
- How to Apply Continuous Probability Concepts to Problem Solving 5:05
- Go to Continuous Probability Distributions

- Go to Sampling

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