Figure 29

The problem of establishing the correspondence between observed differences and mass differences is a part of the weighing method. The first two methods, substitution and transposition, are comparative methods. That is to say, the method requires observations relative to a suitable standard along with the unknown. With these methods, the measurement equipment need be continuous only over the time interval required for making a group of observations and linear only over the range of the difference between the standard and the unknown. Most direct reading equipment is in a sense a substitute standard, that is, at some point in time it is calibrated with reference to a standard, and from that point until recalibration, it is generally assumed to have a long term constancy approaching that of the standard. Most mass measurement equipment can be used either way. The smallest uncertainties invariably will be associated with the comparative mode of operation.

Figure 30

To illustrate the principle, the double substitution method is performed as follows: We start with a simulated equal arm balance, a tare weight — the white cylinder near the base of the balance, a sensitivity weight of known value immediately in front of the dark weight near the center, and two nearly equal brass weights, one with a flat knob in the center and one with a round knob on the left. The scale indication is in arbitrary numbers and the tare weight is necessary to establish an "on scale" condition.

All usual methods result in very similar relations expressing the difference between two objects being compared. In all cases, A minus B is expressed as a ratio between sets of observations multiplied by the value of the sensitivity weight. Obviously requirements for knowledge of the value of m are minimized when the size of the ratio involving the observation is small. The constant of proportionality, K, is really the ratio in front of the bracket terms which we call the value of the division. The strange equal sign is used to indicate that the relations shown are observational equations and not mathematical identities.

Measurement as a Process

A measurement process is essentially a production process, the "product" being numbers, that is, the measurements. A characteristic of a measurement process is that repeated measurements of the same thing result in a series of non-identical numbers. To specify a measurement process involves ascertaining the limiting mean of the process; its variability due to random imperfections in the behavior of the system, that is, its precision; possible extent of systematic errors from known sources, or bias; and overall limits to the uncertainty of independent measurements.

It seems clear that we cannot give an exact answer but will have to content ourselves with a statement that allows for the scatter of the results. Our goal is to make a statement with respect to a new measurement that is independent of all those that have gone before. As indicated in the chart, if we had a sufficiently long record of measurements we could set limits within which we were fairly certain that the next measurement would lie. Such a statement should be based on a collection of independent determinations, each one similar in character to the new observation, that is to say, so that each observation of the collection and also the new observation can be considered as random drawings from the same probability distribution. These conditions will be satisfied if the collection of points is independent, that is free of patterns, trends and so forth; and provided it is from a sufficiently broad set of environmental and operating conditions to allow all the random effects to which the process is subject, to have a chance to exert their influence on the variability. Suitable collections of data can be obtained by incorporating an appropriate measurement into daily routine weighing procedures, for example, a daily measurement of the difference between two laboratory weights, or in the regular calibration of the same weight.

Figure 41

If the measurements tend to cluster when taken close together in time, like the results shown on the chart, some systematic effect is present and certainly the results are not independent. This may be due to some as yet undetermined cause, and the group means may have the appearance of randomness of the previous chart.

Figure 45

Assuming that the limits on the chart are based on large numbers of observations, we would find that very nearly the intended percentage of all such bars, centered on the observed values, would in fact overlap the mean. Only in those cases, such as the points in the area outside of the control limits, will the bar fail to overlap the mean. This is expected in only 1 percent of the cases. More frequent occurrence is a clear indication of either loss of control or that the limits were not properly set. Once we are satisfied that the process has a limiting mean value and is stable enough to permit prediction we turn our attention to evaluating its precision.

Figure 44

We can reverse the process and say that the probability is 99 percent, that the true value, or limiting mean, will not be more than the width of the bar from any observation chosen at random. This will be true of the next observation as well, provided it is an independent measurement from the same process. The probability statement attaches to the sequence of such statements. For each individual new observation the statement is either true or false but in the long run 99 percent of such statements will be true.