No physical quantity can be measured with perfect certainty; there are always errors in any measurement. This means that if we measure some quantity and, then, repeat the measurement, we will almost certainly measure a different value the second time. How, then, can we know the “true” value of a physical quantity? The short answer is that we can’t. However, as we take greater care in our measurements and apply ever more refined experimental methods, we can reduce the errors and, thereby, gain greater confidence that our measurements approximate ever more closely the true value. “Error analysis” is the study of uncertainties in physical measurements, and a complete description of error analysis would require much more time and space than we have in this course. However, by taking the time to learn some basic principles of error analysis, we can: 1) Understand how to measure experimental error, 2) Understand the types and sources of experimental errors, 3) Clearly and correctly report measurements and the uncertainties in those measurements, and 4) Design experimental methods and techniques and improve our measurement skills to reduce experimental errors. Two excellent references on error analysis are: • • John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2d Edition, University Science Books, 1997 Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992

Accuracy and Precision Experimental error is the difference between a measurement and the true value or between two measured values. Experimental error, itself, is measured by its accuracy and precision. Accuracy measures how close a measured value is to the true value or accepted value. Since a true or accepted value for a physical quantity may be unknown, it is sometimes not possible to determine the accuracy of a measurement. Precision measures how closely two or more measurements agree with other. Precision is sometimes referred to as “repeatability” or “reproducibility.” A measurement which is highly reproducible tends to give values which are very close to each other. Figure 1 defines accuracy and precision by analogy to the grouping of arrows in a target.

© G.A. Carlson, 2000 - 2002

Page 1 of 6

Figure 1 – Accuracy and Precision

X XX XX

High precision High accuracy

X X X X X

Low precision High accuracy

X XX XX

High precision Low accuracy

X X

X X X

Low precision Low accuracy

Types and Sources of Experimental Errors When scientists refer to experimental errors, they are not referring to what are commonly called mistakes, blunders, or miscalculations. Sometimes also referred to as “illegitimate,” “human,” or “personal” errors, these types of errors can result from measuring a width when the length should have been measured, or measuring the voltage across the wrong portion of an electrical circuit, or misreading the scale on an instrument, or forgetting to divide the diameter by 2 before calculating the area of a circle with the formula A = π r2. Such errors are surely significant, but they can be eliminated by performing the experiment again C correctly the next time. Experimental errors, on the other hand, are inherent in the measurement process and cannot be eliminated simply by repeating the experiment no matter how carefully. There are two types of experimental errors: systematic errors and random errors. Systematic Errors Systematic errors are errors that affect the accuracy of a measurement. Systematic errors are “one-sided” errors, because, in the absence of other types of errors, repeated measurements yield results that differ from the true or accepted value by the same amount. The accuracy of measurements subject to systematic errors cannot be improved by repeating those measurements. Systematic errors cannot easily be analyzed by statistical analysis....