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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.

If you've been working with z-scores for long, you probably get tired of checking those tables every time you need to check the area under the curve. Luckily, the empirical rule helps us memorize the most important values.

While it may be a bit of an overstatement to say that much of statistics depends on the idea of the normal distribution, it wouldn't be much of one. Without the normal distribution, statisticians would have to perform calculus to find integrals every time they sought to determine how much of a population satisfied a certain requirement. However, due to the fact that, frankly, someone has already done all the hard work for us, much of statistics just involves looking at data and knowing what do to with that data.

However, there are times when simply knowing that there are tables that display all this information simply is not good enough. Without presenting you with simply impossible situations of being stranded on a deserted island with only a set of statistics problems as your own form of escape, the simple fact is that being able to quickly understand what is meant by z-scores of 1, 2, and 3 is valuable. In case you forgot, a **z-score** is just a measure of how many standard deviations something is away from the mean.

The percentages at defined z-scores are always the same. For example, a z-score of 1 will always contain 68% of the individual data points within a set. Likewise, a z-score of 2 will always contain 95%, while a z-score of 3 will always contain 99.7%. As you can imagine, anything beyond a z-score of 3 gets very small.

Just to be clear, in these cases, I'm also including the mirroring z-score of data, so a z-score of 1 includes all data between -1 and 1 standard deviations. If I specifically want to talk about the z-score of 1 without reference to anything on the other side of the curve, I would mention the area under the graph from 0 to 1. As you'd imagine, this is exactly half of what it would be otherwise.

This idea that the vast majority of data will be contained within these three standard deviations is called the **empirical rule**. Note that I'm talking about three standard deviations in both directions. Sometimes you'll have a z-score of -1; that's included as well.

If you've been in the job market or ever looked for an internship, chances are you've probably heard of something called **Six Sigma**, a data-driven approach to process improvement. People claim to have all sorts of colored belts in the stuff, almost like it were a martial art. However, unlike martial arts that take years to fully understand, you can learn the basic meaning of such systems right now.

Like I said earlier, when I mentioned 1 standard deviation, I often mean that whole area from -1 to 1. The difference between -1 and 1 is two. Likewise, the difference between -3 and 3 is 6. Six Sigma seeks to limit errors to occur only outside of those three standard deviations. In order words, it wants to get everything right 99.7 percent of the time.

Let's use the empirical rule to try to figure out just how big a potential customer base is for your company. You've just opened a store in a city of 100,000 people, and your research department forgot to really polish the math down before they gave it to you to present to the board of directors. In total anticipated market size, they only left the z-scores! How would you quickly solve for the total number of anticipated customers?

First off, what is the z-score in question? For the purpose of this exercise, let's say that it's 2. Therefore, your store can be expected to appeal to people two standard deviations above the mean income and two standard deviations below the mean income. Knowing that such an answer is not good enough for your board of directors, you remember that the empirical rule says that such an area of a normal distribution is around 95%. Armed with that, you can confidently claim that your new store will be in a price range that 95% of people can afford.

In this lesson, we looked at the idea of a normal distribution with respect to the **empirical rule**. The empirical rule, also known as the three-sigma rule, states that regular amounts of the total will fall within certain z-scores. Namely, 68% of data will fall between z-scores of -1 and 1, 95% of data will be between -2 and 2, and 99.7% will occur between -3 and 3. This range of 6 standard deviations is the basis of the manufacturing practice of Six Sigma.

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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
- 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
- Estimating Population Percentages from Normal Distributions: The Empirical Rule & Examples 4:41
- 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

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