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Statistics 101: Principles of Statistics11 chapters | 134 lessons | 1 flashcard set

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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.

In this lesson, we will put the normal distribution to work by solving a few practice problems that help us to really master all that the distribution, as well as Z-Scores, have to offer. Review the concepts with a short quiz at the end.

If you've been working with normal distributions for long, you've probably figured out that they are pretty useful things. I'll spare you the pedantic opportunity to roll a pair of dice a thousand times, adding the results of each roll, to show that it is a reliable way of managing large numbers. By now you've also seen the benefits of tools like **standard deviation**, which show you how compact the data set in question is. But wouldn't it be useful to put all that knowledge to use? What if you had to calculate the exact probability of something happening on a normal distribution and only had the standard deviation? Luckily, there's a way to do that.

First, we need to calculate the Z-Score. The **Z-Score** is a measure of how far a point of data is away from the mean in terms of standard deviations. If the standard deviation of a data set is 3, the average is 10, and the point we are looking at has a value of 13, then the Z-score is 1. The formula for that is simple - take the value of the point in question, subtract the mean from it, then divide it by the standard deviation.

*z* = (data point - mean) / standard deviation

Sometimes you'll have negative Z-scores. That is completely understandable. Whereas a **positive Z-score** means that a Z-score represents a value greater than the mean, a **negative Z-score** just means the represented value is less than the mean. For example, a Z-score of 1 represents a value that is one standard deviation greater than the mean, and a Z-score of -1 represents a value that is one standard deviation less than the mean.

With the Z-score you can do something. Oh, by the way, Z-scores are almost always limited to just two decimal points. As you will see, this is because the resulting percentages are very precise.

Before we delve too deep into using the normal distribution, let's be sure that we are squared away on how it works first. The highest point of the curve is where the mean is located. It is often marked with a 0, since the Z-score of the mean is 0.

Now ask yourself if you are going to be dealing with a positive or a negative Z-score. A positive Z-score refers to a standard deviation that is to the right of the mean, meaning that it is greater than the mean. On the other hand, a negative Z-score refers to a standard deviation that is to the left of the mean, meaning that it is less than the mean.

Also, you've got to figure out if you're dealing with a one-sided or a two-sided question, as well as which way the data is going. One-sided questions only ask about one Z-score. Meanwhile, two-sided questions ask about two Z-scores, namely the data in between them or outside of them.

Figuring out which direction the data is going is relatively easy. If it says something like greater or more, you're trying to find the area to the right of the Z-score, while if it says something like lesser or fewer, you're trying to find the area to the left of the Z-score. In this lesson, we're going to learn how to do both greater and less than a certain Z-score.

At the point, it's time to look to the charts. Take a look at the Z-Scores & Standard Normal Curve Areas Table. Exhilarating, I know! Following the columns, find the value for the Z-score in question. For us, we'll look up the Z-score of 2.31. Move down to the 2.3, and then over to the 0.01, and you'll find the percentage for this Z-score. You look it up and find that it is .9896. That means that for a positive Z-score of 2.31, 98.96% of the data is less than 2.31 standard deviations greater than the mean.

If it was a negative Z-score of -2.31, that would mean that 98.96% of the data is greater than 2.31 standard deviations fewer than the mean. In any event, that means that 1.04% of the data is not represented by the set. For a positive Z-score, that would mean 1.04% is greater than 2.31 standard deviations, while for a negative Z-score, 1.04% is less than 2.31 standard deviations.

Again, here's where knowing the difference between a one-sided and a two-sided problem comes in handy. If you are solving a one-sided problem, then you're done! If not, you've got one more step. Luckily, it's pretty easy. If you're looking for the area between two Z-scores, start with the larger Z-score first. Find the area for all data that is less than or to the left of this larger Z-score. Now, find the area value for all of the data that is less than or to the left of the smaller Z-score value. Subtract this from the larger Z-score area to get the area in between. If you are looking for the area beyond two Z-scores, merely subtract that percentage from 1.

That's quite a bit to take in, so let's be sure we understand it by working through a few sample problems.

*Say that you were a theme park insurance agent and wanted to limit access to your park's highest roller coaster. No one under 54 inches is allowed to ride. Average height for your visitors is 66 inches, and standard deviation is 6 inches. What percentage of the population is allowed to ride?*

First, let's find the Z-score.

*z* = (54 - 6) / 6*z* = -2

54 - 66 is -12, which is then divided by our standard deviation, 6, to get the Z-score of -2.

We go to our table for a Z-score of -2.0 and find that the number is .9772. Now ask yourself, are you looking for the numbers that are greater than or less than this Z-score? Obviously, we're looking for greater than, since it's asking how many people above that height can ride. .9772 is greater than 1 - .9772, so we are looking for all the possible points on the right side of the curve. As a result, 97.72% of the population could ride.

That only featured math on one side of the normal distribution - so let's try one that involves it at both sides.

*Imagine that you work for a company that is trying to figure out how many potential customers it has in a city of 100,000 people. Average income is $50,000, and the company knows that people who earn more than $100,000 won't shop at their store, while people who make less than $25,000 can't afford to shop there. The standard deviation for income in this town is $25,000.*

First, calculate the two Z-scores.

*low z* = ($25,000 - $50,000) / 25,000*low z* = 1

For the low end, subtract $50,000 from $25,000 and divide by 25,000. That gives you -1.

*high z* = ($100,000 - $50,000) / 25,000*high z* = 2

Then for the high end, subtract $50,000 from $100,000 and divide by 25,000, leaving a Z-score of 2.

Now, we're looking for the area between the two Z-scores, which represents the number of potential customers for the company. Let's start with the larger Z-score of 2 first. Go to the chart, and you see the value for a 2 is .9772. This means that 97.72% of the population falls to the left of this Z-score.

Look at the smaller Z-score, -1. We find its value in the chart is .8413, but this is a negative Z-score, so remember that this means 84.13% of the population falls to the right of it. To find the area to left, simply subtract it from 1. This gives us 1 - .8413 = .1587, so 15.87% of the population falls to left of this Z-score of -1.

We subtract this value from the larger Z-score value, .9772 - .1587 = .8185, giving us an answer of 81.85%. That means the store could have 81,850 potential customers out of a population of 100,000.

In this lesson we learned how to use the normal distribution in real life to answer questions about large populations. Z-scores give us the ability to calculate the probability of something happening when we're only given the standard deviation.

Remember that a **Z-score** is a measure of how far a point of data is away from the mean in terms of standard deviations. You can use a Z-score chart to find the area value associated with each Z-score under the normal distribution.

Once you have those Z-score areas, you can use them to solve all sorts of problems as long as you think carefully about area. If you're looking for data that is greater than the value represented by the Z-score, you're trying to find the area to the right of the Z-score. While if it says something like lesser or fewer, you're trying to find the area to the left of the Z-score. That's all you need to do for one-sided problems.

Two-sided problems with two Z-scores require one more step. If you're looking for the area between the two Z-scores, find all area for all the data that is less than or to the left of the larger Z-score. Next, find the area value for all of the data that is less than or to the left of the smaller Z-score value. Subtract this from the larger Z-score area to get the area in between. If you are looking for the area beyond two Z-scores, merely subtract that percentage from 1.

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Statistics 101: Principles of Statistics11 chapters | 134 lessons | 1 flashcard set

- 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
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- 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
- Estimating Population Percentages from Normal Distributions: The Empirical Rule & Examples 4:41
- Using the Normal Distribution: Practice Problems 10:32
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