# The infinite experiment and systematic and random errors

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## THEORY

No matter if volume or some other characteristic is measured, a basic understanding of measurements and errors of measurements is essential in scientific work.

Measurements

A measurement can be defined as a way to procure symbols (often numbers) that represent characteristics of an object, an event, or a condition, where the symbols’ relations to each other are the same as the relations between the objects/events/conditions they are representing (Ackoff 1972).

### Classification of measurements

In metrology measurements are traditionally classified into direct, indirect, and combined measurements. The combined measurements have more recently been divided into strictly combined and simultaneous measurements. (Rabinovich 1995) Direct measurements are made by measuring an object with an instrument and reading the results direct on the instrument. Measuring circumferences of limbs with a measuring-tape is an example of direct measuring. The indirect measurements are based on knowledge of the relations between the quantity in interest and other quantities. The other quantities are measured and the wanted quantity is calculated from the results of the measurements. For example can mass and volume be measured to calculate the density of an object, while the ratio of mass and volume is the definition of density. (Rabinovich 1995) Although direct measurements are very common in ortopaedic workshops indirect measurements are probably never performed, at least not manually. CAD/CAM systems may use indirect measurements when calculating volumes of stumps from the scanned coordinates and the approximated shape. Strictly combined and simultaneous measurements are closely related. In both cases several quantities are simultaneously measured (usually direct) and the values are put into a equation system that has to be solved. For strictly combined measurements, the measured quantities are of the same type. The equation system the results are put into is based on the relationship between arbitrary objects with the same measurable quantity. Assume that the mass of an object is known. The mass of several different objects can then be found by comparing different combinations of objects, and constructing equations from the results. For the simultaneous measurements, on the other hand, the measured quantities are of different kinds and the equations reflect relationships between the quantities in the nature. (Rabinovich 1995) Assume that the relationship between pressure and temperature of a gas is studied. The coefficient describing the relationship can be found by measuring the pressure by different temperatures and solving the equation system formed by the results. Strictly combined and simultaneously measurements are probably never performed in orthopaedic workshops.The strictly combined measurements can be viewed as a generalisation of the direct measurements and the simultaneous measurements as a generalisation of the indirect measurements. This means that, when it comes to the measurements physical significance, they can only be classified into direct and indirect measurements. Still, when the processing of the data after the measuring is in focus, it is practical to distinguish between (a) direct, (b) indirect, and (c) simultaneous and combined measurements. (Rabinovich 1995) Ackoff (1972) classifies measurements in an alternative way and take example of four types of measurements: numbering, counting, ranking, and measuring in a “restricted” sense. When numbering, symbols (letters, numbers, et cetera) are put on objects or events and the symbols are used as identification of the objects/events in the further processing of the data. No arithmetic operations can be performed with the symbols. When counting a number of positive values are put on elements in a class. For example can money of different denomination be counted: 20+0,5+10 = 30,5. The numbers resulting from the counting can be used in all kinds of arithmetic operations. The elements of interest can also be ranked, which means that the elements are put in a specific order depending on a specific relationship between the elements. Although numbers can be used to describe an element’s rank, no arithmetic operations can be performed with the numbers. Finally, there is also a measurement in a restricted sense. These measurements are made with a constant measuring unit. (Ackoff 1972) .

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INTRODUCTION
THEORY
Measurements
Classification of measurements
Scales of measurement
Errors of measurement
The infinite experiment and systematic and random errors
Propagation of error
Direct measurements
Indirect measurements
Simultaneous and combined measurements
Determination and correction of errors
Measurements and errors in orthopaedic workshops
MATERIAL AND METHODS
Study design
Volume determination
Determination of volume using circumferential measurements
Determination of volume using CAPOD
Accuracy
Theoretical accuracy of tip volume
Theoretical accuracy of stump volume
Practical accuracy of stump volume
Precision and sources of error
Measuring procedure
Intrarater precision
Interrater precision
Sources of error
Subjects
Statistical methods
RESULTS
Accuracy
Theoretical accuracy of tip volume
Theoretical accuracy of stump volume
Practical accuracy of stump volume
Precision
Intrarater precision
Interrater precision
Sources of error
DISCUSSION
Main results
Accuracy
Theoretical accuracy of tip volume
Theoretical accuracy of stump volume
Practical accuracy of stump volume
Precision
Intrarater precision
Interrater precision
Sources of error
Clinical implications
Methodological considerations and limitations of the study
External accuracy
Internal accuracy
Other methodological considerations
Future studies .
CONCLUSIONS
ACKNOWLEDGEMENTS
REFERENCES
APPENDIX I, INSTRUCTIONS FOR THE MEASUREMENTS
APPENDIX II, MEASURING FORM
APPENDIX III, CALCULATIONS
Error limits for cut cone
Error limits for sphere segment

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