Lesson 3: 2.3 Density Curves and the Normal Distribution
Duration of Days: 4
Lesson Objective
Students will be able to describe and interpret a density curve, locate the mean and median on a density curve, and understand the relationship between the area under the curve and the proportion of data.
How do we transition from a "bumpy" histogram of a sample to a "smooth" model of a population?
Why must the total area under a density curve always be exactly 1 (or 100%)?
On a skewed density curve, why does the mean get "pulled" further toward the tail than the median?
Density Curve
Mean of a Density Curve
Median of a Density Curve
Uniform Distribution
HSS-ID.A.1: Represent data with plots on the real number line.
HSS-ID.A.2: Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets.
SAT "Data Inferences" require students to understand that a curve represents a population. Knowing where the mean and median sit relative to each other on a skewed curve is a common conceptual question on the test.
To move students from "Sample Thinking" to "Population Modeling." It introduces the notation mu and sigma for the first time, distinguishing them from the sample statistics x and s they used in Chapter 1.
The Problem: A random number generator provides a number between 0 and 5. The density curve for this distribution is a horizontal line (a uniform distribution).
Part A: What is the height of this density curve? (Hint: Area must be 1).
Part B: What percent of the time will the generator give a number between 1 and 3.5?
Height vs. Area: Students often think the height of the curve at a specific point is the "probability." It’s vital to reinforce that only the area over an interval represents a proportion.
The Mean/Median "Tug of War": Students often forget which one moves more in a skewed distribution. Use the "balancing" analogy—the mean is the pivot point and must move toward the "heavy" tail to keep the curve from tipping.
Support: Use physical cut-outs of a skewed density curve made of cardboard. Have students try to balance the shape on their finger to find the mean.
Extension: Have students find the density curve height for a triangle-shaped distribution (e.g., a distribution from 0 to 2 where the peak is at 0) and calculate proportions using the formula for the area of a triangle.
Teacher assigns examples from the textbook and other resources.
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