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The Bell Curve


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Abstract

The goal of this lesson is to introduce the concepts of the normal curve, skewness and the standard deviation. The controversy over the 1994 book is also examined.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to the normal distribution
  • have an understanding of how the normal distribution has been used and misused to support conclusions

Student Prerequisites

  • Arithmetic: Student must be able to:
    • Students should understand sums, differences, and quotients for all activities.
  • Geometric: Students must be able to:
    • Students will need to understand both the concept of area in general and, specifically, the area of a rectangle.
  • Technology: Students must be able to:
    • Each student or group of students working together will need a computer with a Java-capable browser. Students should be comfortable using the computer and browser. Calculators may be helpful for solving problems that arise in discussions.

Teacher Preparation

Key Terms

bell curveSee normal distribution
meanThe sum of a list of numbers, divided by the total number of numbers in the list. Also called arithmetic mean
median"Middle value" of a list. The smallest number such that at least half the numbers in the list are no greater than it. If the list has an odd number of entries, the median is the middle entry in the list after sorting the list into increasing order. If the list has an even number of entries, the median is equal to the sum of the two middle (after sorting) numbers divided by two. The median can be estimated from a histogram by finding the smallest number such that the area under the histogram to the left of that number is 50%
normal distributionAlso called "bell curve," the normal distribution is the curved shape of a graph that is highest in the middle and lowest on the sides
standard deviationStandard deviation tells how spread out numbers are from the average, calculated by taking the square root of the arithmetic average of the squares of the deviations from the mean in a frequency distribution

Lesson Outline

  1. Focus and Review

    Have students review key terms and concepts related to this lesson including:

    • Area of a rectangle
    • Various representations of data
    • Mean, median, mode, and range
    • Percentages

  2. Objectives

    Let students know what they will be learning and doing today. Say something like this:

    • Today we are going to be looking at one specific way that mathematicians and statisticians look at data: the bell curve. We will be examining what this representation is and the controversy behind it.
    • We will be using the computer and Internet later, but for now I want you to keep the computer closed.

  3. Teacher Input

    Lead a discussion on the "Bell Curve" controversy.

  4. Guided Practice

    Introduce the concept of standard deviation or spread with the normal distribution activity. Also explore the differences between individual data points, samples of various sizes, and expected theoretical distributions.

  5. Independent Practice

    Revisit "Bell Curve" with a discussion.

    Revisit the ideas of mean, median, mode, and standard deviation graphically with the Skewness activity.

  6. Closure

    Interconnect ideas of theoretical and experimental probability with discrete-valued histograms and continuous distributions.

Alternate Outline

  • Look at the normal distribution without discussing the bell curve controversy. This saves time and avoids controversial material.
  • Relate the histograms produced by the normal and skew distributions to the idea of finding area under a curve by counting the rectangles that fit under it (simple graphical numerical integration). Use the grid and the scale on the normal distribution to find the area under the normal curve. (It should be close to 1).

Suggested Follow-Up

This lesson introduced the notion of the normal curve and illustrated a real life (mis)use of the statistics drawn from a given situation. Revisiting the initial lessons on Statistics and Shopping and Probability and Sports and applying the new knowledge on statistics would make an excellent capstone activity.

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