Reporting the Results of an Experimental MeasurementExample© G.A. Carlson, 2000 - 2002 Page 1 of 6Experimental Errors and UncertaintyNo physical quantity can be measured with perfect certainty; there are always errors inany measurement. This means that if we measure some quantity and, then, repeat themeasurement, 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 experimentalmethods, we can reduce the errors and, thereby, gain greater confidence that our measurementsapproximate ever more closely the true value.“Error analysis” is the study of uncertainties in physical measurements, and a completedescription of error analysis would require much more time and space than we have in thiscourse. 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 thosemeasurements, and4) Design experimental methods and techniques and improve our measurement skills toreduce experimental errors.Two excellent references on error analysis are:• John R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties inPhysical Measurements, 2d Edition, University Science Books, 1997• Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis forthe Physical Sciences, 2d Edition, WCB/McGraw-Hill, 1992Accuracy and PrecisionExperimental error is the difference between a measurement and the true value orbetween two measured values. Experimental error, itself, is measured by its accuracy andprecision. 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 notpossible to determine the accuracy of a measurement.Precision measures how closely two or more measurements agree with other. Precisionis sometimes referred to as “repeatability” or “reproducibility.” A measurement which is highlyreproducible 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 2 of 6Figure 1 – Accuracy and PrecisionTypes and Sources of Experimental ErrorsWhen scientists refer to experimental errors, they are not referring to what are commonlycalled 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 thelength should have been measured, or measuring the voltage across the wrong portion of anelectrical circuit, or misreading the scale on an instrument, or forgetting to divide the diameter by2 before calculating the area of a circle with the formula A = π r2. Such errors are surelysignificant, but they can be eliminated by performing the experiment again C correctly the nexttime. Experimental errors, on the other hand, are inherent in the measurement process andcannot be eliminated simply by repeating the experiment no matter how carefully. There are twotypes of experimental errors: systematic errors and random errors. Systematic ErrorsSystematic errors are errors that affect the accuracy of a measurement. Systematic errorsare “one-sided” errors, because, in the absence of other types of errors, repeated measurementsyield results that differ from the true or accepted value by the same amount. The accuracy ofmeasurements subject to systematic errors cannot be improved by repeating those measurements.Systematic errors cannot easily be analyzed by statistical analysis. Systematic errors can bedifficult to detect, but once detected can be reduced only by refining the measurement method ortechnique. Common sources of systematic errors are faulty calibration of measuring instruments,poorly maintained instruments, or faulty reading of instruments by the user. A common form ofthis last source of systematic error is called “parallax error,” which results from the user readingan instrument at an angle resulting in a reading which is consistently high or consistently low.High precisionHigh accuracyLow precisionHigh accuracyLow precisionLow accuracyHigh precisionLow accuracyXXXXXXXXXXXXXXXXXXXX© G.A. Carlson, 2000 - 2002 Page 3 of 6Random ErrorsRandom errors are errors that affect the precision of a measurement. Random errors are“two-sided” errors, because, in the absence of other types of errors, repeated measurements yieldresults that fluctuate above and below the true or accepted value. Measurements subject torandom errors differ from each other due to random, unpredictable variations in the measurementprocess. The precision of measurements subject to random errors can be improved by repeatingthose measurements. Random errors are easily analyzed by statistical analysis. Random errorscan be easily detected, but can be reduced by repeating the measurement or by refining themeasurement method or technique.Common sources of random errors are problems estimating a quantity that lies betweenthe graduations (the lines) on an instrument and the inability to read an instrument because thereading fluctuates during the measurement.Calculating Experimental ErrorWhen a scientist reports the results of an experiment, the report must describe theaccuracy and precision of the experimental measurements. Some common ways to describeaccuracy and precision are described below. Significant FiguresThe least significant digit in a measurement depends on the smallest unit which can bemeasured using the measuring instrument. The precision of a measurement can then beestimated by the number of significant digits with which the measurement is reported. Ingeneral, any measurement is reported to a precision equal to 1/10 of the smallest graduation onthe measuring instrument, and the precision of the measurement is said to be 1/10 of the smallestgraduation.For example, a measurement of length using a meterstick with 1-mm graduations will bereported with a precision of ±0.1 mm. A measurement of volume using a graduated cylinderwith 1-ml graduations will be reported with a precision of ±0.1 ml.Digital instruments