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

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  • 0:01 Normal Distributions
  • 0:32 Estimating Percentages
  • 1:04 The Empirical Rule
  • 2:10 Real-Life Applications
  • 3:01 Forecasting a Customer Base
  • 3:59 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

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.

Normal Distributions

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.

Estimating Percentages

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 Empirical Rule

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.

Real-Life Applications

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.

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