Now that I had that most difficult data display under my belt I was prepared for the next data set, Figure Twelve.
It was a series of points forming the quadrant of a circle, with dispersion in the data. I assumed that the center of the circle was the intersection of a horizontal line from the top data point of Course #67 and a vertical line through the two points of Course #86. The latter two points were not part of the circle; they provided a reference for the vertical pointer.
To obtain this quadrant I had to use linear values for both the vertical and horizontal scales, with course thickness in cubits and course number. This was a departure from the previous data plots which were all scaled with vertical logarithmic values and pyramid height.
The graphical plot showed that a circle centered on the intersection of those two lines did, indeed, define this set of data.
However, I had to shrink the vertical axis to make a circle. My original plot formed an ellipse. Given the previous data set, and how well it was defined by delimiter lines and pointers I was naturally curious about the possible shrinking of scale. Here there were no visible delimiters. I could alter the scale for visual display as I pleased.
Fortunately, the designer used other elements to reassure us of our solution.
The wide dispersion of the data is similar to that found in Data Set One.
The visual dispersion is conditioned by the vertical scale, whether graphed as an ellipse or a circle. Note that we could also "flatten" the scale vertically to make the elongation of the ellipse in the horizontal direction, rather than stretch it to make the elongation vertical.
To test the validity of the display I felt I was being invited to calculate the dispersion as I had done on the first three data sets. However, I could not do so without going to polar coordinates. I had to redefine the positions of the data points according to circular function.
Rather than define in terms of radii and angles, I calculated using the method of squares, except for courses #74, #75, #84, and #86, which are clearly not part of the circle. The mean of the calculated values is shown by the darker circle. Please note that I could have accomplished the same results by using polar coordinates. Actually, the method of squares is equivalent.
We can see that the designer was showing his finesse with a data set once again. He scattered the data in such a way that we are forced to calculate a three-sigma band limit based on polar coordinates. I show this with the concentric dashed circles.
From Data Sets #1 through #3 we know he could control dispersion to suit his display needs.
We should give note to the fact that the designer is here displaying finesse with a set of data defining circular mathematics, while he showed similar finesse with the exponential mathematics in the previous data displays.
We can see how the designer "boxed in" the data points with three-sigma band limits. The paired sets at courses #68 to #71 fall on either side of the mean. Courses #76 and #77 have opposite pair points that fall on (or nearly) the three-sigma band limits, while their mates fall near the mean. Other oddities can be noted.
The curiosity of the three-sigma bands is their difference from the calculated mean. They are both exactly, within error, equal to two horizontal course numbers.
This fact led me to believe that my solution of a circle was true, rather than a solution of an ellipse. If I had stretched or shrunk the axis to display ellipses I would have equally stretched or shrunk the three-sigma bands at the contrasting vertical or horizontal center positions. The dispersion bands would then not have been equal distance from the mean.
From the graph it would appear that the dispersions bands might be slightly smaller in radius than those I show. However, the circular properties modify the calculations somewhat from the visual evidence.
The data points at courses #74 and #75, and the two at #84 and #86 were a puzzle. Why place them outside the main distribution? Note how the two at #74 and #75 are arranged to fit on another concentric circle with a diameter again two units (course numbers) less than the inner three-sigma band. As I mentioned above the pair at #86 served to define the vertical radius of the circle. The pair at #84 served to get our attention because it is separated from the pair at #86 by two units, the value of the three-sigma band limits. The separation of the pairs at #74 and #75 was one unit, or one course.
We cannot compare this dispersion with Data Set #1 because of the difference in scales, from vertical logarithmic thickness and horizontal pyramid height to vertical linear thickness and course number.
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