## The Famous Bell Curve or Normal/Gaussian Probability Distribution

The Normal Distribution is the most important probability distribution in statistics and probability theory. Another name for the Normal Distribution is the Gaussian Distribution, named in honor of the great German mathematician Carl Friedrich Gauss (1777 – 1855) who used the Normal Distribution to explain the apparent errors in the observed locations of stars. Because the shape of the Normal curve is bell shaped, it is also called the Bell Curve.  However, there are many other probability distributions, such as the Student’s t-distribution, that have a bell shape curve.

The purpose of this post is to show how I introduce the Normal Distribution and its amazing properties to my students after they have learned how to calculate means and standard deviations of relatively small populations and samples of numerical data. Do not expect to see mathematical proofs of the various properties of the Normal Distribution, an exhaustive description of its many applications, nor a discussion of the discovery and history of the Normal Distribution. In passing, I should mention that Gauss used the Normal Distribution, but he did not discover it.

Shown below is a short list of some applications of the Normal Distribution.

• Describe characteristics of human populations such as distribution of IQ scores, heights, body weights, body temperatures, blood pressure, school achievement test results, and human mortality.
• Errors in measurement and errors/differences between experimental and predicted results.
• Size and variability of parts produced in a manufacturing process.
• Approximate other probability distributions such as the Binomial and Poisson.
• Approximate the mean or average value of a population.
• Approximate what percent of a population has some characteristic of interest such as what percent of the population favors a particular political candidate

The two most equally important parameters, numerical characteristics, of any population, not just a Normal population, is the population mean µ and the population standard deviation σ. Parameter µ equals the mean or average value of the population, and σ tells us how much the numerical data values of the population are spread out. As σ increases, the more spread out the population data values are. From a quality control point of view, a primary objective of a manufacturing process is to keep the σ’s of the parts dimensions as small as possible.

For the Normal Distribution, µ and σ completely determine all properties of the distribution. In fact, µ and σ appear in the somewhat complicated equation of the Normal curve. Unless a curious student asks me the equation of the Normal curve, I don’t bother to show my students the equation. I hope some readers will be curious enough to Google the equation.

The slide show below shows three normal curves where µ = 100 for all three curves, and σ = 16, 8, or 4. As σ decreases, the curves become less spread out. I kept the same x-y axes scale for all three graphs in order to reinforce what standard deviation σ is all about.   A remarkable property of all Normal curves is that for any µ and any σ, the area under the curve from µσ to µ + σ always equals 0.6827. Another remarkable property of all Normal curves is that the area under the curve from -∞ to +∞ = 1. The proof of this fact requires a clever integration technique that is not taught in lower level calculus courses. Students can easily verify this fact with a graphing calculator by finding the area under a Normal curve from -9999999 to 9999999. The TI-84 graphing calculator function calls normalcdf(-9999999, 9999999, 12, 3) and normalcdf(-9999999, 9999999, 0, 1) both return a value equal to 1. I use my program Probability Simulations to create graphic images in real time similar to the graphic images in this post. As computer screen images are projected on a dropdown screen, students are required to use their graphing calculators to verify the resulting numerical calculations shown on the projection screen. I find this helps students better understand the concepts presented in the lesson.

The next two groups of slides shows the area under normal curves from µ – kσ to µ + kσ for various values of µ, σ, and k. For every µ, σ, and any fixed constant k > 0, the area under a Normal curve from µ – kσ to µ + kσ is always the same. When k =1, 2, and 3, these areas when rounded to 4 decimal places equal 0.6827, 0.9545, and 0.9973 respectively. I tell my students that it is imperative that they memorize and understand the significance of 68%, 95%, and 99.7%. Areas under continuous probability distribution curves correspond to probabilities of events in a sample space or the percent of a population that falls in an interval of interest. X-Y axes scale values were set so that the shape of the curves is the familiar plumb bell shape. I also tell my students that if the graph of a Normal curve is magnified 1,000,000 times, they will see a gap between the curve and the horizontal axis even though the curve appears to touch the horizontal axes in drawings of the curve.

The slides in the first group describe the population of times in minutes that it takes Super Lube to service a car. I remind students that a Normal population is the set of all possible numerical data values of interest where the distribution of the data values is described by the Normal Distribution.   The next three slides describe the population of reading rates in words per minute of sixth-grade students. A quick inspection of only the graph in the first slide immediately tells us that about 68% of sixth-grade students have a reading rate ranging from 101 to 149 words per minute.   Now let’s take a look at a typical exercise relating to the Normal Distribution. I tell my students to always draw a sketch of the graph of the Normal Distributed described in the problem if there is no graph provided. A good graph of the distribution enables a person with a trained eye to immediately extract a great deal of information with just a quick inspection of the graph. Learning how to draw a good sketch of a Normal curve given µ and σ helps develop the idea that the horizontal axes of a Normal curve is a statistical measuring stick where σ is the fundamental unit of length. The statement of the problem along with solutions is given below. Solutions were found with a TI-84 graphing calculator and checked with my Probability Simulations program.

Problem: The Quick Burger drive-through restaurant chain is concerned about the amount of time customers spend waiting in the drive-through. A recent study of Quick Burger waiting times found that customer waiting times are approximately normally distributed with mean µ = 138.5 seconds and standard deviation σ = 29 seconds. Answer the following questions. Questions: (Round probabilities to 3 significant digits.)

1. What is the probability that a randomly selected customer will wait in the drive-through less than 100 seconds?
2. What percent of customers will wait in the drive-through more than 3 minutes?
3. What proportion or percent of the customers spend between 2 and 3 minutes in the drive-through?
4. What is the probability that a customer will spend exactly 138.5 seconds in the drive-through?
5. Use the 2σ rule to determine what customer wait would be considered unusually long?
6. What is the probability that a randomly selected customer would have an unusually long wait time?
7. What is the probability that a randomly selected customer would have an unusually short wait time?
8. Quick Burger is considering rewarding a customer with a \$5.00 coupon if the customer’s wait time is too long. However, Quick Burger does not want to reward more than 0.1% of its customers. What should be the cut off wait time for a \$5.00 coupon? (Round up to the nearest minute.)

Solutions: The next set of eight slides shows the solutions of the above questions. When I discuss the solution of a question with my students, I project the shaded area under a curve on my computer screen to a drop down screen in the classroom, and have my students use their TI graphing calculator to verify the probabilities shown on the screen. Initially some students struggle with entering function calls into their calculator, but with a little practice they get the hang of it.        In closing, I will mention that readers can download my free handout Summary of Common Probability Distributions and my free Windows program Probability Simulations by going to www.mathteachersresource.com. I can’t imagine teaching statistics or probability without the Probability Simulations program and my TI-84 graphing calculator which has revolutionized the way statistics is taught.