scope:

Introduction

ASTM work often requires the presentation of observed data on two variables. The objective of the presentation generally is to show the nature and extent of the relationship between the variables and the uses to which the relationship may be used. Pairs of measurements may exhibit a linear or a nonlinear relationship. The treatment of nonlinear relationships is beyond the scope of this manual. However, it is often possible to transform nonlinearly related measurements by some suitable mathematical process so that a linear, or approximately linear, relationship is assumed by the data. Some of the commonly used transformations are the logarithms of the data, their roots, reciprocals, or other suitable functions. A discussion of transformations is given in Section 11.

It is usually advantageous to plot paired data so that the existence and nature of any relationship between them may be more readily observed. There are several ways of drawing a straight line through the plotted points. The simplest is by eye. When the scatter of the points is slight the line so drawn may be quite adequate for most purposes. For example, if experimental results showing the relationship between proportional limit and tensile strength of dental alloys are as shown in Fig. 1, a statistical analysis is probably unnecessary. Similarly, if experimental data calibrating a new method of determining calcium in the presence of large amounts of magnesium are as shown in Fig. 2, the analysis is very straight-forward. Different people observing these data would probably draw the same graph, and furthermore, such plots would be representative of the true relationship between the variables. On the other hand, when the scatter of the points is considerable, visual plotting is too uncertain; different lines may be drawn by different individuals, thereby leading to nonobjectivity in the results. An example of such data is found in Table I and is plotted in Fig. 3. These data are taken from an actual experiment (l)¹ where a statistical analysis was carried out on certain mechanical properties of cast and wrought gold dental alloys. Several relationships were found between the mechanical properties, with the relationship between proportional limit and tensile strength shown in Fig. 3. If quantitative use is to be made of these data, visual plotting is unsatisfactory, and a more objective procedure is necessary.

As a further example where visual plotting is inadequate, suppose the data obtained for calibration of a new method of determining calcium in the presence of large amounts of magnesium are as shown in Table II and as plotted in Fig. 4. Ten different samples containing known amounts of calcium oxide were analyzed by the new method. Here again it is evident that a more objective method of determining the relationship between two variables is needed. The method used should be such that not only will all observers derive the same relationship from a given set of data, but it should also describe how the data can be used to predict results about future observations. The following sections will deal with these problems.

¹ The boldface numbers in parentheses refer to the list of references appended to this paper.