A key element of the Plan is to decide how to measure the selected response and explanatory variates on the units in the sample. To determine the value of any variate on a unit, we call the measuring devices, methods and individuals involved the measuring process. Once a measuring process is specified, it is important to understand its properties. We call measurement error the difference between the value of the variate determined by the measuring process and the ``true'' value. Measurement error is propogated through the Analysis and hence to the Conclusions.
In many applications, a separate smaller PPDAC cycle is carried out to investigate the attributes of the measuring process within the overall study. We define the properties of the measuring process in terms of repeatedly measuring the same study unit. Two concepts are measuring bias, an attribute of the (target) measuring process describing systematic measurement error, and measuring variability, an attribute of the (target) measuring process describing the change in the measurement error from one determination to the next.
Michelson paid careful attention to the measuring processes he had specified for his study and discussed at great length investigations he undertook to ensure that there was little measuring bias and variability. Consider, for example, the measurement of the distance between the two mirrors [39](page 125). To avoid bias, he calibrated a steel tape against a Wurdeman copy of the standard yard. The calibration used a comparator with two microscopes, one fixed and one that can be moved towards or away from the fixed microscope by turning a screw. The distance between the microscopes was set to 1 standard yard. Then the tape was placed in the comparator so that .1 ft corresponded to the cross-hairs of the fixed microscope and the length of the first yard of the tape was determined by rotating the screw until the cross-hairs of the movable microscope corresponded to 3.1 ft on the tape. This procedure was repeated 33 times to determine the cumulative number of turns of the screw corresponding to the length of the tape from .1 ft to 99.1 ft. The temperature was recorded so that an adjustment (unexplained) could be made.
Next, he carried out a separate study to determine the distance corresponding to 1 turn of the screw of the movable microscope. This was accomplished by measuring 20 times the number of turns that correspond to 1 mm and then averaging. It is clear that Michelson appreciated the power of averaging to reduce variability in measurement. Combining the results of the two studies and adjusting for temperature, the corrected length of the 100 ft steel tape was 100.006 ft.
To measure the distance between the two mirrors (approximately 2000 ft), the plan was to place lead markers along the ground and use the tape to measure the distance from one to the next following a carefully defined standard procedure. The tape was to be placed along the (nearly) level ground and stretched using a constant weight of 10 lbs. This led Michelson to investigate the stretch of the tape.
To adjust for stretch, another small study was conducted in which the tape was stretched using a 15 lb force and the stretch in mm at 20 ft intervals was measured. The data are shown below.
| Length | Amount of Stretch |
| 100 | 8.0 |
| 80 | 5.0 |
| 60 | 5.0 |
| 40 | 3.5 |
| 20 | 1.5 |
In the language we have introduced, for this small study, the study population using a 15 lb force is different from the target population which requires a 10 lb stretching force. Note also the curious weighted average for estimating the amount of stretch per foot of tape.
The goal of introducing the corrections for stretch and length of the tape was to reduce bias in the final measurement of the distance between the two mirrors. To reduce the variability of the distance measurement, the procedure was repeated 5 times (with corrections for temperature on each). The temperature corrected measurements varied from 1984.93 to 1985.17 ft. Michelson used the average of the 5 determinations and then corrected for stretch and bias in the tape to get his final measure of distance between the two mirrors.
The case study is an excellent example of a careful scientist reducing measurement error from his measuring processes using two different approaches. Based on empirical studies, he reduced bias by calibration and correction, and he reduced variability by averaging. At the conclusion of his paper, Michelson provided a detailed discussion of the effects of possible measurement bias on his estimate of the speed of light. It is alarming to realize how often modern data are produced and analyzed with little consideration for the properties of the measuring process. 55