Tracking Change Conditions
With Phase Planes

A method to better understand the interaction between two variables

by D. Keith Denton

Editor's note: The Toolbox for the Mind, written by D. Keith Denton and published by ASQ Quality Press (item H1011), explains how to approach and solve problems using innovative techniques. The following is an excerpt explaining how to determine and analyze the degree of change in two variables over periods of time.

A phase plane is a visual tool describing the change condition of any system by identifying its "current location on a common two-dimensional Cartesian plane."1

The data plotted results in a pattern of interactions that can be analyzed to predict reasonable expectations and solve a number of problems that would be hard to visualize otherwise. Business ratios are a good source of such data as these ratios can be plotted to show changes in production efficiency, quality control and any common financial ratio.

Dissecting and plotting a phase plane

Phase planes plot changes in two measures. They plot the differences between each measure value and its value in a previous time period. Changes can be either positive or negative from one period to another. The center of the phase plane is coordinate 0, 0 and represents no change in either measure.

Phase planes are divided into four quadrants (see Figure 1). Each quadrant represents specific behaviors for any two interactive measures (income to sales, for example). Quadrants are traditionally numbered counterclock-wise starting with the upper right quadrant.

Because phase planes show the interaction of two variables, each axis is used to plot the changes in one of the measures. Points on the phase plane represent the relationship between these two measures (X and Y) of change: The change in one type of defect (numerator) to the change in total defects (denominator). The lines connecting these points track the points as they move through time--the changes that occur within a system.

Points on a phase plane are calculated by comparing the changes in value from one period to the next. For example, if 20 total defects (X variable) were detected in the first time period and 23 total defects in the second period, the marginal difference would be +3. Similarly, if one particular defect (Y variable)--the diameter of a hole--had eight instances in the first period and 10 instances in the second period, the difference would be +2. These differences become the point that is plotted (+3, +2). A trajectory is produced by drawing a line to each succeeding point (see Figure 1).

Understanding patterns

Over time, data should accumulate in more than one quadrant. Data in quadrants 1 and 3 indicate near proportional changes between any defect and total defects. Quadrant 2 data shows that a particular defect is increasing, while total defects are declining. If the data is mainly in quadrant 4, the measured defect is declining, while other defects are increasing.

Two-point trajectories, such as that in Figure 1, are a common way for related data to oscillate. For instance, production to income and cycle time to sales often yield a two-point trajectory as both points go up (diameter defects up, total defects in process up) and down together. The things causing one problem also affect the other.

A four-period oscillation occurs whenever one measure is oscillating or changing twice as fast as the other. It tends to create a slightly out of sync figure of eight.

More complex oscillations are possible when a great number of assignable causes that are unique to one variable and not the other exist.

Random fluctuations that are all over the grid exemplify a situation in chaos.


1. Richard H. Priesmeyer, and Edward G. Cole, "Phase Plane Analysis: A Nonlinear Way to Track Quality," a presentation given at the Third Annual Chaos in Manufacturing Conference, 1995.

D. KEITH DENTON is a professor and distinguished scholar at Southwest Missouri State University-Springfield. He has a doctorate in education from Southern Illinois University in Carbondale.

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