Normal Distributions Math 728 Maria Coca
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Contents 1. Class .................................................................... 1 2. Introduction ..........................................................2 3. References ............................................................. 3 4. Learning Objectives ...............................................3 5. Materials and Keywords .............................................3 6. General Outline .................................. 3 (a) Lesson Presentation ............................................3 (b) Worksheet 1: Empirical Rule ....................................5 (c) Worksheet 2: Standardized Normal distribution and z-scores ....6 (d) Lesson Summary ................................................8 7. Extensions............................................................10 8. Teacher Reflection ...............................................10
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Class
This lesson was designed for a Statistics and Probability class. For this lesson students must be familiar with standard deviation, area under the curve, arithmetic mean, z-scores as well as their mathematical symbols. Throughout this lesson the teacher will lead the discussion of normal distributions on the board and students will alternate between working in groups of maximum three students and taking notes/class participation. In addition, this lesson was designed for 90 minutes length.
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Introduction
The normal distribution is also known as the Gaussian distribution, and in France as Laplace’s distribution. Gauss (1777-1855) and Laplace (1749-1827) brought out the central role of normal distribution in the theory of errors of observation Quetelet (1796-1874) and Galton (1822-1911) fitted the normal distribution to empirical data such as heights and weights in human and animal populations. But the normal distribution was actually first discovered around 1720 by Abraham De Moivre (1667-1754), as the approximation to the binomial (n, p) distribution for large n [1]. The appearance of the normal distribution in many contexts is explained by the central limit theorem. According to this result, for independent random variables with the same distribution and finite variance, as n → ∞, the distribution of the standardized sum (or average) of n variables approaches the standard normal distribution [2]. It can be shown that this happens no matter what the common distribution of the random variables summed or averaged, discrete or continuous, provided the distribution has finite variance. In particular, the central limit theorem implies that the distribution of the sum or average of a large number of independent measurements will typically tend to follow the normal curve, even if the distribution of individual measurements does not. This mathematical fact is the basis for most statistical applications of the normal distribution. Finally, the normal distribution is often fitted to an empirical distribution of observations. the parameter µ and σ are usually estimated by the mean and standard deviation of the list of observations. Examples of the kinds of observations where the normal approximation has been found to be good are weighings on a chemical balance, and measurements of angular position of a star [1].
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References 1. Pitman, J. P robability. Springer, 93-98. 1992. 2. Starnes, D., Yates, D., Moore, D. StatisticsT hroughApplications. WH Freeman. ll109-131. 2004.
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Learning Objectives
Students will: • Learn the fundamental properties of Normal distributions • Familiarize with standard distributions • Use standard distributions to compare a variety of results derived from Normal distributions • Understand how large sets of data can be studied using probabilistic distribution approaches 2
• Estimate percentages and understand the general concepts of probability theory • Recognize over and under estimations, and identify better approximations
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Materials Required • work sheets • scratch paper, pencil
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Keywords
Normal distribution, Standard normal distribution, Mean, Standard deviation, area under the curve, chance outcome.
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General Outline: 1. Do Now: Introduction to the normal curve. 2. Introduction to density curve (the area under this curve approximates “chance outcomes”). 3. Quick note on their relevance and real world applications. 4. Historical note about normal distributions. 5. Standard normal curve i.e. the 68-95-99.7 rule, equivalent to 34-14-2 on the x-scale axis. 6. Standard normal distributions (µ = 0, σ = 1). Standardizing a normal distribution . Here z measures the number of standard deviations from the mean µ using z = x−µ σ to the number x, then we say that z is x in standard units.
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Lesson Presentation
1. Students will be requested to do a “Do Now”: On the board the teacher will draw a normal curve and will ask students what they see, and as the students share their impressions, the teacher annotates on the board her/his students observations. This engaging activity can be exercised for 5 minutes. 2. Now, the teacher can introduce students to a formal definition. A lesson hand out should be provided prior to formal definition (see attached lesson notes). It is suggested that a student read the formal definition at loud followed by the teacher’s explanation. Definition 1. The normal distribution with mean µ and standard deviation σ is the distribution over the x-axis define by areas under the normal curve with these parameters (µ, σ) [1]. 3
Explanation : The curve has two parameters the mean µ and the standard deviation σ.Here µ can be any real number positive or negative, while σ can be any strictly positive number. The mean µ indicates where the curve is located, while the standard deviation σ marks a horizontal scale. Think of the normal curve as continuous histogram, defining a “chance outcome” distribution (Probability distribution) over the line by relative areas under the curve. Then µ indicates the general location of the distribution, while σ measures how spread out the distribution is. So that the total area under the curve is 1 or 100 percent. 3. The teacher throughout hers/his explanation must emphasize the characteristics of Normal curves: symmetric, single-peaked, and bell-shaped curves. Their tails fall off quickly, so outliers are not expected. In addition, since Normal distributions are symmetric the mean and median lie together at the peak in the center of the curve [2]. 4. Now, students get to practice the definition by an activity titled “Representing Data I: Using Frequency Graphs.” This document can be found at the end of this lesson plan and at: http://map.mathshell.org/materials/lessons.php?taskid=404subpage=concept This activity involves students working in small groups of 2 to 3 students to match frequency graphs with with possible interpretations. This activity will engage students to test their understanding of normal curves.Time estimated for this activity 20 minutes. 5. The teacher will gather the class for a collective class participation regarding the students answers to the activity, then introduce the class to the second activity,that is explaining that the Normal (Density) curve describes a continuous distribution of chance outcomes over a number line. Moreover, the teacher will give a brief history note about the normal distribution. 6. On the lesson hand out students will have a plot of a normal curve that describes the empirical rule, and clearly indicates the mean µ, and standard deviation σ. 7. Students will solve worksheet 1: Empirical Rule (68-95-99.7), in groups of 2 students and will be given 20 minutes to complete this activity. 8. The teacher will gather the class back for solutions and reflections about the type of problems that can be solve using normal distribution. 9. Finally, one of the students will draw her/his solution on the board from worksheet 1. Then the teacher will describe that the standard normal curve is the normal distribution with µ = 0 and σ = 1. In other words, the standard normal distribution is the distribution on the standard unit or z − scale. At this point students will be asked to solve worksheet 2: z-scores in groups of 2 students (different from worksheet 1 group activity). Students will have 15 minutes for this activity. 10. The teacher will gather the class for discussion one last time. this time students will summarized how will they describe a normal curve and the most important characteristics as well as a real world application. 4
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Worksheet 1
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Empirical Rule (68-95-99.7)
1. The final scores in a large class on Principles of Statistics are normally distributed with mean score of 60 (µ = 60) and standard deviation of 10 (σ = 10). (a) Sketch the normal curve that represents the distribution of the final scores, and clearly indicate its mean (µ) and standard deviation (σ).
(b) If the lowest passing score is 50, what proportion of the class is failing?
(c) If the highest 80% of the class is to pass, give an estimation of the lowest passing score?
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2. The heights of male freshmen entering a large state university are normally distributed with a mean of 68 inches (µ = 68 inches). It is know that about 2% of the freshmen are taller than 72 inches. (a) Sketch a normal curve that represents the distribution of heights and clearly indicate its mean.
(b) What is the standard deviation of distribution of heights?
(c) About what percent of the freshmen males at this college have a height of 66 inches or less? Sketch a normal curve and clearly indicate the corresponding shaded region that illustrates your solution.
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Worksheet 2 z-scores
1. Write the formula for computing standardized scores (z-scores):
2. Jacob scores 16 on the ACT. Emily scores 670 on the SAT. The ACT scores for more than 1 million students in the same class were roughly normal with mean of 20.8 and standard deviation of 4.8. The SAT scores for 1.4 million students in a recent graduating class were roughly normal with a mean of 1026 and standard deviation of 209. (a) Compute the z-scores for Jacob and Emily.
(b) Sketch the two standardized normal curves for Jacob and Emily.
(c) Assuming that both tests measure scholastic aptitude, who has the higher score? Sketch a standardized normal curve that indicates your solution.
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Lesson Summary
1. Normal distribution is the distribution used most in daily life to model real world situations. This distribution is the most important distribution that you will learn in this year. 2. It is known that Normal distribution was first used by Carl Friedrich Gauss, so it is sometimes called the Gaussian distribution. 3. The Normal distribution is a symmetric distribution that is centered around a mean and spreads out in both directions. 4. A Normal curve is completely described by its mean and standard deviation.
Figure 1: Normal distribution for different σ values
Examples : 1. SAT, ACT scores. 2. The heights of all female students at JJSE. 3. Sale purchases for 500 customers at Target. 4. Test scores for all Introduction to Statistics courses.
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Empirical Rule
The empirical rule is described by the following percentages 68 − 95 − 99.7 rule, where 68% of the distribution falls within one standard deviation from the mean. 95% of the distribution falls within two standard deviation from the mean. 99.7% of the distribution falls within three standard deviations from the mean.
Figure 2: Empirical Rule 68-95-99.7 rule
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Standard Normal Distribution
All normal distributions are the same if we measure in units of size σ about the mean µ as center. Changing to these units is called standardizing. To compute this standardization we use the following formula: x−µ z= σ The standardized value z is called z-score and tells us how many standard deviations away from the mean a particular value in the distribution is.
Figure 3: Standardizing Normal distributions
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Extensions
After a week of practice normal distributions, students should be introduced to the standard normal density curve: 1 1 2 x−µ φ(z) = √ e 2 z , where z = σ 2π In addition, this lesson can be extended to the standard normal cumulative distribution function (c.d.f)by graphically demonstrating the relationship between the x-scale from the normal curve to the z − scale standard curve. For this lesson, students would have been introduced to indefinite integrals.
Figure 4: Normal distribution and c.d.f for different σ values Wikipedia images
The standard normal c.d.f gives the area to the left of z under the standard normal curve.
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Teacher Reflection
This lesson was already executed. Students found this lesson to be challenging because it covered several new topics in a short period of time. It was suggested to break the lesson in two parts, one for normal distributions and another for standardized normal distribution. Overall, students seemed to enjoy learning new vocabulary and real world applications regarding their class work.
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