Griffith Feeney 03-29-96 HAND PLOTTING TECHNIQUE INTRODUCTION Plotting and visualization are fundamental to data analysis, and hand plotting provides both a useful skill and background understanding for use of computer plotting tools. REFERENCE The basic reference on hand plotting technique is John W. Tukey's Exploratory Data Analysis (Addison-Wesley Publishing Company, Reading, Massachusetts, 1977). Look up 'plotting' in the index and read the referenced portions of the book. MATERIALS Materials required for plotting are several different kinds of graph paper for use as templates (10 by 10 rules to the half inch is a good basic paper), drafting paper (Clearprint No. 1000HP recommended), and a few paper clips to fasten drafting paper to graph paper. EQUIPMENT Plotting should generally be done in pencil, and mechanical pencils are best. The lead diameter should not be too small, and the lead should be sufficiently soft to make easily visible marks. A good eraser, such as MagicRub, is essential. A small electric erasor, such as the Sakura Electric, is a great convenience, and nearly essential for certain fine work. A flat, hard, clean work surface is of course essential. PLOT ON TRACING PAPER, NOT ON GRAPH PAPER Tukey explains why. To begin a plot, secure a fresh piece of tracing paper to a piece of graph paper. Read the rules on the graph paper through the tracing paper and plot accordingly. It is often convenient to mark up a piece of graph paper as a template to be used repeatedly. Various templates may be made for various sizes and types of plot. USE SCALE TRANSFORMATIONS TO CONTROL SIZE AND ASPECT RATIO Sometimes the ranges of the values to be plotted will mesh neatly with the scales available on the graph paper. If it does not, however, scale transformations should be used to control the size and aspect ratio of the plot. The ASPECT RATIO of a plot is the ratio of the vertical plotting range in physical units (inches, centimeters, etc.) to the horizontal plotting range in the same units. A square plot has aspect ratio 1. Scale transformations are extra work, which we aim to minimize. For most line plots we can usually manage with a scale transformation for the vertical axis only. For scatter plots, however, it is generally necessary to use transformations for both axes. MECHANICS OF SCALE TRANSFORMATONS These are best illustrated by example. Suppose that we are using 10 by 10 to the half inch graph paper and that we want a six inch square plotting area. We begin by making a template by drawing a six inch square on a piece of graph paper. The horizontal scale will have 120 rules (6 inches times 20 rules per inch). We make tick marks and scale values at lines 0, 10, 20, ..., 120 on both the left and right sides of the square. Tick marks should extend out from (not into) the square so as not to intrude in the plotting area. Putting scales on the right as well as the left makes it easier to plot point to the right and reduces the chance of careless errors. Tracing paper is then secured to this template in preparation for plotting. Suppose now that we are plotting a time series of total fertility rates and that a suitable range for the values to be plotted is 2 to 6 children per woman. We need to transform the observed scale [2,6] to the plotting scale [0,120]. Think of this as a transformation of intervals. We begin with [2,6], subtract 2 to get [0,4], divide by 4 to get [0,1], and multiply by 120 to get [0,120]. To plot the observed value y, then, we first compute 120 times (y - 2)/4 and plot the resulting value on the tracing paper using the plotting scale on the graph paper template. The arithmetic may be simplified, of course, but it is useful to envision every scale transformation problem as a matter of first getting from the observed range to [0,1] and then from [0,1] to the plotting range. It is sometimes useful (see exercise) to use plotting scales that begin with something other than zero. In such cases there is one additional step USE OF PROGRAMMABLE CALCULATORS Tukey recommends writing down transformed values before plotting to avoid careless errors. This may be avoided by the use of a programmable calculator, into which the scale transformation may be programmed. This represents a great saving of time and effort and is highly recommended. It does of course require one to have a programmable calculator and to know how to program it. Where both abcissa and ordinate values must be transformed, a little programming work will enable the user to key in the two values to be transformed, separated by a suitable delimiter, and obtain the y and x plotting coordinates out in the form 'y.x'. This requires rounding the plotting coordinate for y to the nearest integral value and adding to it the plotting coordinate for x divided by (in the case of 3 digit plot scale values) 1000. EXERCISES 1. Select four different sets of 6-10 numbers whose plots you would like to compare (e.g., parity distributions for four countries, or for four different points in time for a single country). Define a rectangular plotting area, divide it into four equal sized sub-plotting 'windows', and use scale transformations to draw the four plots into the four windows. 2. Describe the procedure for inverting a plot scale transformation. The inverse transformation allows one wants to go from a plotted point on a given scale to the corresponding observed value. This is used to secure numerical values from data available only on a plot, e.g., published plots with no numeric data or hand smoothed/interpolated values from plotted data.