A Geometry Teacher Meets the Correlation Coefficient

Steve Phelps, Jul 16, 2015

An investigation into the correlation coefficient and some unexpected conics

Table of Contents

  1. The Problem That Started it All
    1. The Original Problem
  2. Correlation Fun Facts
    1. The Correlation Coefficient as a Slope
    2. Correlation as the Cosine of an Angle
    3. Correlation as a Geometric Mean of Slopes
    4. Correlation as the Mean of Rectangle Areas
  3. Fun Fact About Regression Lines
    1. Correlation and the Least Squares Regression Line
    2. Finding Equations of LSRL: Old School
  4. Fun Facts About Z-Scores
    1. Z-Scores as Geometric Transformations
  5. A Simpler Problem
    1. Correlation of the Vertices of a Triangle
    2. Guess the Equation

1.

The Problem That Started it All

  • 1. The Original Problem

1.1.

The Original Problem

This was the original problem posed to my students. The points A, B, C, D, and E are [b]strongly associated[/b] (why?) yet have a [b]correlation = 0[/b] (why?) Slowly drag point A and find other possible locations for point A that keeps the correlation = 0. By careful dragging, you should be able to find many of them. When you find such locations, click on the [color=#0a971e]Capture Point[/color] button. When you have captured many of these points, make a conjecture as to where these points lie.
– Steve Phelps

2.

Correlation Fun Facts

  • 1. The Correlation Coefficient as a Slope
  • 2. Correlation as the Cosine of an Angle
  • 3. Correlation as a Geometric Mean of Slopes
  • 4. Correlation as the Mean of Rectangle Areas

2.1.

The Correlation Coefficient as a Slope

Before we explore the original problem further, let's examine some other interpretations of the correlation coefficient. In the applet below, points A, B, C, D, and E have the correlation shown and are best modeled by the [b]blue least squares regression line[/b].[color=#1551b5][/color]. Points A', B', C', D', and E' are the [b]Standardized Points[/b] (the coordinates are z-scores). The equation of the LSRL that best fits these points is a line through the origin (a direct variation) with a slope equal to the correlation coefficient.
– Steve Phelps

2.2.

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2.3.

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2.4.

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3.

Fun Fact About Regression Lines

  • 1. Correlation and the Least Squares Regression Line
  • 2. Finding Equations of LSRL: Old School

3.1.

Correlation and the Least Squares Regression Line

Correlation is closely related to the equation of the [b]Least Squares Regression Line (LSRL)[/b]. Why is it called the Least Squares Regression Line?[br][br]Drag the two "Drag Me!" points until the sum of the squares of the residuals is as small as possible. Once you are satisfied that you can not[br]make the sum of squares any smaller, click on the check boxes to see the true LSRL and the centroid of the points A, B, C, D, and E. [br][br]One property of the LSRL is that it passes through the centroid of the points, of the point of the mean coordinates.
Correlation and the Least Squares Regression Line
– Steve Phelps

3.2.

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4.

Fun Facts About Z-Scores

  • 1. Z-Scores as Geometric Transformations

4.1.

Z-Scores as Geometric Transformations

What if you explored the idea of standardizing coordinates as a geometric transformation? Explore angle measures and areas and perimeters. Computing a z-score (standardizing) is [math]z = \frac{x - \bar{x}}{s}[/math], a composition of a translation and a dilation.
– Steve Phelps

5.

A Simpler Problem

  • 1. Correlation of the Vertices of a Triangle
  • 2. Guess the Equation

5.1.

Correlation of the Vertices of a Triangle

The figure below is a simpler version of the original problem posed to my students. The vertices of Triangle ABC are my points if interest. Drag point A slowly in order to find other locations for point A such that the correlation of points A, B, and C remains zero. When you find a point, click on the [color=#0a971e]Capture Point[/color] button. I have included the Standardized Triangle as well. What do you think is true about this triangle when the correlation is zero? What patterns do you notice about all possible locations of point A that produce a correlation of zero?
– Steve Phelps

5.2.

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