Geometric Aid in Teaching Rectilinear Correlation

Abstract

THE PURPOSE of this paper is to present a simple geometric aid for use in the teaching of rectilinear correlation. The coefficient of correlation may be concerned with the relationship between a number of variables. In the materials to follow, the concern will be with just two variables, X and Y. A c o n v e nient means of indicating the concept of correlation is by a diagram in which the X variable is plotted on the horizontal axis and the Y variable on the ver tical axis. If the points within the axes scatter in an upward direction from left to right, in a manner which indicates that X and Y tend to vary together, a positive correlation is present. If the scatter is downward from left to right, a negative correla tion is present, indicating that X and Y tend to vary inversely. If there appears to be no trend pres ent, then zero correlation is indicated. A convenient way of teaching rectilinear corre lation is to consider the scattering present as de scribing an ellipse. An ellipse is completely de fined by its major and minor axes. The m a j or axis is the long axis, and the minor axis is the short axis. An ellipse varies in shape with two interesting limiting cases. One limit occurs when the major and minor axes are equal; then the fig ure is a circle. The other extreme is when the minor axis is equal to zero and the major axis is finite, in which case we have a straight line. First, plot the points on the correlation chart, or scatter diagram. Thus one obtains an approx imate elliptical figure by drawing a curve which encloses most points and passes through some of the outer points. Next draw a single straight line through the long axis of the figure. This line rep resents the trend of the scattering points. For descriptive purposes this can be handled quickly by inspection of the scatter and crudely drawing a line by hand. (A more precise method would be to draw the lines representing the regression of Y on X and X on Y, and then to use a line which is intermediate between the two to represent the trend. ) This line would be the major axis of the ellipse. The minor axis would be a line drawn perpendicular to this line at its center. Having drawn the scatteringof the points to represent an ellipse, we then determine the mag nitude of t h e relationship. This magnitude could be determined exactly by an appropriate formula available. The approximate magnitude, however, could be ascertained by looking at the ratio of the minor axis to the major axis, provided the two axes are drawn to the same scale. As the minor axis goes to zero (or the major axis gets very large relative to the minor axis), this ratio will also go to zero, and the correlation coefficient will vary toward one. When the minor axis equals zero and the major axis is finite, the ratio also equals zero, the coefficient is one, no spread is present, and we have a single straight line?with perfect corre lation. (An exception occurs when the slope of the line representing the major axis equals zero. In this case, the Y variable equals a constant value for every X, and the line is parallel to the X axis. Thus, there is no change in Y with a change in X; Y remains the same. In this case, even though the minor axis is zero and the ratio is zero, the correlation coefficient would be zero because no