Statistics – Math Teacher's Resource Blog

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 µ – σ 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.)

What is the probability that a randomly selected customer will wait in the drive-through less than 100 seconds?
What percent of customers will wait in the drive-through more than 3 minutes?
What proportion or percent of the customers spend between 2 and 3 minutes in the drive-through?
What is the probability that a customer will spend exactly 138.5 seconds in the drive-through?
Use the 2σ rule to determine what customer wait would be considered unusually long?
What is the probability that a randomly selected customer would have an unusually long wait time?
What is the probability that a randomly selected customer would have an unusually short wait time?
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.

Release of Probability Simulations Software by Math Teacher’s Resource

I’m happy and excited to release a free full Windows version of the software Probability Simulations. The primary reasons for writing the program are: 1) Satisfy my lingering curiosity about a variety of probability distributions. 2) Develop an effective classroom software tool that makes it easy to demonstrate a variety of concepts and interesting problems in probability theory and statistics.

Here is a brief description of the software’s main features:

The software provides a total of 23 probability simulations ranging from classic geometric probability simulations to common probability distributions such as the binomial, Poisson, normal, and chi-square.
Upon selection of a particular simulation, the purpose of the simulation is explained and the key parameters of the simulation are described. Simulation parameters can be easily edited.
A simulation can be run in either manual or automatic mode which gives users two different ways to experience a simulation, view program output and interact with the software. In manual mode, the user is required to click the <Run Simulation> button or press the <R> key to run the next simulation.
Users can run simulations of sampling statistics based on populations described by probability distributions such as binomial, normal, student-t, chi-square and uniform continuous.
Users can run simulations of confidence interval calculations or simulations of hypothesis tests from a P-value point of view or traditional rejection region point of view.
Experimental and expected probabilities of events can be found by just moving the cursor over a histogram bar and then shift-click the mouse.
Probabilities, inverse probabilities, critical values and P-values of statistics can be found by just clicking a button or shift-clicking the mouse in the region under a probability curve.
The software is easy to use because of the program’s <Help> menu command and context sensitive explanations are provided at the appropriate moment in all simulations.

The screen shots below give you a quick peek at some of the program’s features.

Click here to learn more about the software and download it for free. Since the program is copyright protected, users are bound by the terms stated in the license agreement. I invite teachers to experiment and play with the software. I’m sure that teachers of subjects other than statistics will find some of the simulations interesting and useful in their own classes. Let me know if you would like to see additional features in future versions of the software and how you use the software in your classes. Thanks!

Also check-out my free Summary of Common Probability Distributions, which describes the key properties of probability distributions found in lower level college statistics courses.

Binomial Probability Distribution and the Battle of Gettysburg

Joshua Chamberlain (1828-1914). See question #11 in the Gettysburg Trivia Quiz.

This post shows how I use my Battle of Gettysburg Trivia Quiz to help teach the binomial probability distribution. My brother Roger and I are Civil War buffs, and we recently paid another of our many visits to the Gettysburg Military Park Battlefield in Gettysburg, Pennsylvania. On each visit to a Civil War battlefield, I’m rewarded with another gold nugget of information about a particular battle that relatively few people know. After several visits to the Gettysburg battlefield, I created a Gettysburg Trivia Quiz to demonstrate the binomial probability distribution in my statistics classes at the junior college where I have taught a variety of math courses over the years. I consider the binomial probability distribution to be the most important discrete probability distribution in statistics. (The normal probability distribution is the most important probability distribution in all of statistics.)

You can download the Gettysburg Trivia Quiz by clicking the links below. I believe you’ll find it fun, interesting and helpful to take the quiz before continuing.

Gettysburg Trivia Quiz (student version)

Gettysburg Trivia Quiz (teacher version)

You can also download the quiz on our free instructional content page under the statistics tab.

The features of the quiz and how I administer the quiz are as follows:

The quiz has 20 multiple choice questions.
Each question has 5 choices.
Each question is designed so that no normal person would have any idea of what the correct answer is. Hence a normal test taker can only guess at the correct answer, and therefore has a 0.2 probability of answering any question correctly.
All questions are independent of each other in that knowledge of any one question can’t help answer any other quiz question.
On the day of the lesson, I announce that the class will be taking 20-question multiple choice trivia quiz on the battle of Gettysburg.
I distribute a copy of the quiz to each student, and tell them there is no reason to take a peek at a neighbor’s answer because their quiz score has no influence on their course grade.
I tell them that the intent of the quiz is to eventually help them better understand the binomial probability distribution. Most likely there will be no high scores.
I allow 5-10 minutes to take the quiz. (You will observe intense expressions on some faces.)
I verbally provide the correct answers so that each student can grade his/her own quiz. I ask students to be honest when grading their quiz. There should be no surprise if almost all students have a very low score.
As each student reports his/her quiz score, I record the score on the chalkboard.
I then begin a discussion of the binomial probability distribution, introduced previously, and how the binomial distribution can be used to predict the probability distribution of scores from the Gettysburg Trivia Quiz. The class will see that the expected average or mean score is 4, and an unusually high score is 8 or more. A score of 8 is more than 2 standard deviations above the expected mean score of 4, where the expected standard deviation of the scores equals 1.789.
Using our calculators, the probability of a score of 0 = 0.0115, probability of a score of 4 = 0.218, and the probability of a score of 8 or more = 0.0321.
We then compare the experimental binomial results from the quiz with the expected binomial results. Especially with a larger class, students are amazed at how the expected and experimental results agree.

My free handout, Summary of Common Probability Distributions, describes the key properties and some applications of the probability distributions found in lower level college statistics courses. You can also download the handout on our free instructional content page under the statistics tab. Teachers can copy parts or all of the handout to share with students. Here is a general description of the key components of a binomial probability distribution:

A Bernoulli trial is any experiment that has exactly two possible outcomes. Examples: 1) Tossing a single coin has outcomes of head or tail. 2) Answering a multiple choice test question has outcomes of correct answer or incorrect answer. 3) Consider the chances of a person living to retirement age of 65. That person can die before 65, or age 65 or after.
The two possible outcomes of a Bernoulli trial are called ‘success’ or ‘failure’. The terms success and failure are just labels that represent the two possible outcomes of a Bernoulli trial.
Each outcome of a Bernoulli trial has no influence on subsequent Bernoulli trials. In other words, Bernoulli trials are independent events.
The random variable x of a binomial population equals the number of successes found in n Bernoulli trials repeated under identical conditions. Therefore x could equal any of the integers that range from 0 to n.
The probability of success on each trial is always the same and is denoted by p. The probability of failure on each trial is denoted by q. The laws of probability tell us that p + q = 1 or q = 1 – p.
The expected average or mean of random variable x equals μ = np.
The expected standard deviation of random variable x equals σ = √(n*p*q).
The formula for the probability of exactly x successes in n trials is P(x) = _nC_x * p^x * q^(n-x).

Now let’s see how the binomial probability distribution can be used to predict the key statistical properties of the sample of quiz scores. Note that the number of test takers has no influence of the expected values for μ and σ below. As the number test takers increases, we can expect increased agreement between expected and experimental results.

Each quiz question is a Bernoulli trial; the answer is correct or incorrect.
The trivia quiz is composed of 20 independent Bernoulli trials with p = 0.2 and q = 0.8 because we are assuming that normal students don’t know any of the details of the battle of Gettysburg.
The number of successive independent Bernoulli trials equals n = 20.
Let the random variable x equal the number of correct answers reported by test takers.
The expected mean or expected class average score μ = np = 20*0.2 = 4 correct answers.
The expected standard deviation of quiz scores σ = √(npq) = √(20*0.2*0.8) = 1.789.
An unusually high score = μ + 2σ = 4 + 2*1.789 = 7.58 = a score of 8 or more.
Calculate the sample mean x-bar and sample standard deviation s of the reported scores, and then compare these values with μ and σ.
P(x = 0) = 0.0115, P(x = 4) = 0.218), and P(x ≥ 8) = 0.0321. Compare these probabilities to the experimental probabilities derived from the reported quiz scores. Example: What is the experimental probability that a student got a score of 4?

Based on the Gettysburg Trivia Quiz, here are the results of two probability simulations of a binomial distribution where the number of Bernoulli trials n = 20 and the probability of success on each trial p = 0.2. The widths of the histograms bars = 1 unit. Both the heights of the blue bars and the area of the blue bars equals P(x) = the expected probability of a student having exactly x successes in 20 Bernoulli trials. The heights of the red bars and the area of the red bars equals the experimental probability of a student having exactly x successes in 20 Bernoulli trials. The first graph shows the results of 50 students taking the trivia quiz, and the second graph shows the results of 50,000 students taking the quiz. Every simulation run on any number quiz takers will give us different experimental results, but the same expected results.

Using the same simulation parameters for the two graphs above, the graph below shows the cumulative results after a run of 30 simulations in manual mode where the user is required to press the <R> key to run the next simulation. The output of each simulation in manual mode shows us how well a test taker did on the quiz. The graph below includes a string of S’s and F’s that indicate successes and failures for test taker number 30 in 20 Bernoulli trials. The probability simulation software used to create the histograms allows me to find expected and experimental probabilities of events by just moving the mouse cursor over a histogram bar, and then shift-left clicking the mouse. The software also makes it easy to find probabilities such as P(x ≤ 5) or P(x ≥ 2).

My Probability Simulations software was used to create the graphics in this post. I will be releasing a full and free version of this software in the near future at the Math Teacher’s Resource website. The software release will be announced in a future post. Stay tuned!

I will close this post by telling you a true story about Larry (not his real name) who was in a general education high school math class that I taught approximately 20 years ago. Larry had very little mathematical ability. Larry could learn how to solve a specific type of math problem when the problem was presented in a specific format. If the presentation of a math problem was modified in any way, Larry was lost.

Like myself, Larry was a Civil War buff. Many of his classmates thought Larry was a history genius because he could spout a wide variety of American history facts. Larry could tell you that Private Hugh White was the lone British sentry on guard duty near the British Customs House on Monday, March 5th, 1770, the night of the Boston Massacre. I loved to share Civil War stories with Larry. He liked to ask trivia questions to stump me. It turns out, Larry, his father and I also had the same barber, Joe. About 6 years after Larry graduated, I asked Joe to have Larry take my Gettysburg Trivia Quiz the next time he got a haircut. I gave Joe the answer key so that he could grade the quiz while Larry was still in the shop. I told Joe that if Larry scored 19/20 or 20/20 on the quiz, I would pay for his haircut. Larry scored 19/20! The haircut and tip were on me. Joe told me how excited Larry became, and how proud Larry’s father was. I guess our brains are wired differently.