By a graph we mean a representation of numerical values or functions by the positions of points and lines on a two-dimensional surface. A graph is inherently more limited in precision than a table of numerical values or an analytic equation, but it can contribute a "feel'' for the behavior of data and functions that numerical tables and equations cannot. A graph reveals much more clearly such features as linearity or nonlinearity, maxima and minima, points of inflection, etc. Graphical methods of smoothing data and of differentiation and integration are sometimes easier than numerical methods. Graphs and graphical methods suffer, of course, from the limitations of a two-dimensional surface. Thus, normally, a plotted point has only two degrees of freedom, which we assume here to be represented by one independent variable x (increasing from left to right) and one dependent variable y (increasing from bottom to top). The word "normally" is required in the above sentence because three-dimensional plots, representing, for example, points with coordinates x, y, and z, can be represented in oblique projection on a two-dimensional surface by computer techniques.
Wherever possible, the data from an experiment should be plotted at an early stage, even if numerical methods will be used subsequently for greater accuracy. This is particu-larly true when functions are to be fitted by least-squares or other methods, since a graph may make evident special problems or requirements that might otherwise be missed.
Following are sections on:
A. Manual preparation of graphs
For constructing a graph by hand, the following six step procedure is recommended.
Plan it so as to make best use of the area available and present the data in a manner that will be clearest to the viewer. In most cases the entire useful area of a normal-sized sheet of graph paper should be dedicated to the graph. In some cases, the paper should be turned sideways (i.e., with the binding margin on top) so that the independent variable, x, runs longways on the page, to provide an aspect ratio more favorable for displaying the desired features.
Determine the maximum and minimum values of x and y (remember, it is not necessary that they both be zero at the lower left corner!) so as to fit the points, lines, and curves into the available area. Be sure to make adequate allowance of space for borders all around, scale numbers and scale labels, at least at the bottom and at the left, and a legend telling what the graph is about.
The graph paper should be of good quality, accurately ruled with thin, lightweight lines. In most cases these will constitute a grid of lines equally spaced in two perpendicular directions; special papers are available for plotting logarithmic functions in one direction ("semilog paper") or in both directions ('log-log paper"), or for plotting in polar coordinates, etc. Most good graph papers are ruled in colors that are soft enough that they do not distract the viewer from the plotted lines and points, which are usually in black. The most commonly used graph papers are ruled with a 1-mm spacing, the lines at 5 and 10 mm being heavier. The smallest divisions on the paper should preferably represent multiples of 1 or 2 or perhaps 5 in the plotted variable.
Draw, with heavy pencil lines, a rectangle within which all points and lines will be plotted, leaving sufficient margins all around, extra width for the binding margin, and also adequate allowance for the scale numbers and legends mentioned above. Indicate the major scale divisions with short, heavy pencil marks and label them with appropriate numerical values. Under the scale numbers for the independent variable enter the scale label, stating the quantity that is varying, and its units, e.g.,
for absolute temperature in kelvin. To avoid scale numbers that are too large or too small for convenient use, multiply the quantity by a power of 10, e.g.,
for the density of a gas. This is equivalent to multiplying all the scale numbers by 10-4 but is much more convenient. Similarly, enter the scale label for the dependent variable along the side scale on the left; if necessary, turn the paper sideways to write the label so that it will run vertically.
4. Plot the Points; Draw the Lines and Curves.
Using a sharp, hard pencil, plot the experimental points as small dots, as accurately as possible. Try to plot with an accuracy of one-fifth of the smallest grid division. Draw small circles (or squares, triangles, etc.) of uniform size (2 or 3 mm) around the points in ink to give them greater visibility.
If it is desired to draw a straight line to provide a linear fit to the plotted points, draw a faint pencil line with a good straightedge such as a transparent ruler or draftsman's triangle, and do not hesitate to erase the line and try again. When you are satisfied that you have made the best possible visual fit, go over the line again with a softer pencil or a pen to make it sufficiently heavy; this time, do not allow the line to cross the circles or other symbols drawn around the experimental points. Smooth nonlinear curves may be drawn either freehand or with ships curves, splines, or other devices, with as much trial and error and erasing as required.
If a "theoretical curve" is to be drawn based on an analytic function representing predicted behavior, plot the function at as many points, closely spaced, as reasonably possible; do not draw symbols around them. Carefully draw a smooth curve through the points, lightly at first, and then "heavy it up" as desired.
Somewhere on the graph (if possible, at the bottom) enter a legend with an identifying figure number. The legend should state what the graph is about, identify symbols .and line types used, and provide all needed information that will not be provided in the document to which the graph will be attached.
6. Check the Overall Appearance.
After all points, lines, curves, scales, and legends are in place, check the overall clarity and legibility of the graph. If points or lines are too faint, make them heavier; if the graph is messy or confusing, attempt to improve it by erasing and redrawing the features concerned. If the end result falls seriously short in the expected qualities, start over. Your second attempt will probably take much less time to prepare, and it will look much better and will better suit its intended purpose.
In most cases, you should take advantage of the ability of computer spreadsheet programs to generate graphs. These may be generated on the computer's monitor screen for inspection and editing and then printed out on ink-jet or laser printers. Much of the above discussion of manual preparation of graphs applies also to computer graphs. Since computer graphs can be produced quickly, it may be worth while to make several tries to obtain the best results. The first plot may be a "default plot'' to see the overall appearance of the plotted data; then user-chosen scales, symbols, aspect ratio, and labels may be introduced as needed. If the computer output uses generic labels such as x and y, be sure to relabel with appropriate variables.
It frequently happens that an intermediate or final result of the calculations in a given experiment is obtained from the slope or intercept of a straight line on a graph - for example, a plot of y against x. In such a case it is desirable to evaluate the uncertainty in the slope or in the position of the intercept. A rough procedure for doing this is based on drawing a rectangle with width 2D(xi) and height 2D(yi) around each experimental point (xi, yi), with the point at its center.
The significance of the rectangle is that any point contained in it represents a possible position of the ''true'' point (xi, yi) and all points outside are ruled out as possible positions. Having already drawn the best straight line through the experimental points, and having derived from this line the slope or intercept, now draw two other (dashed) lines representing maximum and minimum values of the slope or intercept, consistent with the requirement that both lines pass through every rectangle. as shown in the figure below. Where there are a dozen or more experimental points, it may be justifiable to neglect partially or completely one or more obviously "bad'' points in drawing the original straight line and the limiting lines, provided that good judgment is exercised. The difference between the two slopes or intercepts of the limiting lines can be taken as an estimate of twice the limit of error in the slope or intercept of the best straight line.
Where the number of points is sufficiently large, the limits of error of the position of plotted points can be inferred from their scatter. Thus, an upper bound and a lower bound can be drawn, and the lines of limiting slope drawn so as to lie within these bounds. Such graphical methods are justifiable only for rough estimates. In either case, the possibility of systematic error should be kept in mind.