Name Section Date Lab Partner(s) EXPERIMENT 1 Experimental Uncertainty (Error) and Data Analysis Advance Study Assignment Read the experiment and answer the following questions. 1. Do experimental measurements give the true value of a physical quantity? Explain. 2. Distinguish between random (statistical) error and systematic error. Give an example of each. 3. What is the difference between determinate and indeterminate errors? 4. What is the difference between measurement accuracy and precision? Explain the general dependence of these properties on the various types of errors. (continued) 1 EXPERIMENT 1 Advance Study Assignment 5. What determines how many fi gures are signifi cant in reported measurement values? What would be the effect of reporting more or fewer fi gures or digits than are signifi cant? 6. In expressing experimental error or uncertainty, when should (a) experimental error and (b) percent difference be used? 7. How could the function y 5 3t2 1 4 be plotted on a Cartesian graph to produce a straight line? What would be the numerical values of the slope and intercept of the line? 2 EXPERIMENT 1 Experimental Uncertainty (Error) and Data Analysis INTRODUCTION AND OBJECTIVES methods of error and data analysis that may be used in subsequent experiments. Laboratory investigations involve taking measurements of After performing the experiment and analyzing the physical quantities, and the process of taking any measure- data, you should be able to do the following: ment always involves some experimental uncertainty or error.* Suppose you and another person independently took 1. Categorize the types of experimental uncertainty several measurements of the length of an object. It is highly (error), and explain how they may be reduced. unlikely that you both would come up with exactly the same 2. Distinguish between measurement accuracy and pre- results. Or you may be experimentally verifying the value of cision, and understand how they may be improved a known quantity and want to express uncertainty, perhaps experimentally. on a graph. Therefore, questions such as the following arise: 3. Defi ne the term least count and explain the meaning and importance of significant figures (or digits) in • Whose data are better, or how does one express reporting measurement values. the degree of uncertainty or error in experimental 4. Express experimental results and uncertainty in appro- measurements? priate numerical values so that someone reading your • How do you compare your experimental result with report will have an estimate of the reliability of the an accepted value? data. • How does one graphically analyze and report 5. Represent measurement data in graphical form so as to experimental data? illustrate experimental data and uncertainty visually. In this introductory study experiment, types of experi- *Although experimental uncertainty is more descriptive, the term error mental uncertainties will be examined, along with some is commonly used synonymously. EQUIPMENT NEEDED • Pencil and ruler • Hand calculator • Rod or other linear object less than 1 m in length • 3 sheets of Cartesian graph paper • Four meter-long measuring sticks with calibrations • French curve (optional) of meter, decimeter, centimeter, and millimeter, † respectively †A 4-sided meter stick with calibrations on each side is commercially available from PASCO Scientifi c. THEORY 1. Unpredictable fluctuations in temperature or line voltage. A. Types of Experimental Uncertainty 2. Mechanical vibrations of an experimental setup. Experimental uncertainty (error) generally can be 3. Unbiased estimates of measurement readings by the classifi ed as being of two types: (1) random or statistical observer. error and (2) systematic error. These are also referred to as Repeated measurements with random errors give slightly (1) indeterminate error and (2) determinate error, respec- different values each time. The effect of random errors tively. Let’s take a closer look at each type of experimental may be reduced and minimized by improving and refi ning uncertainty. experimental techniques. Random (Indeterminate) or Statistical Error Random errors result from unknown and unpredictable Systematic (Determinate) Errors variations that arise in all experimental measurement situa- Systematic errors are associated with particular measure- tions. The term indeterminate refers to the fact that there is ment instruments or techniques, such as an improperly no way to determine the magnitude or sign (+, too large; –, calibrated instrument or bias on the part of the observer. too small) of the error in any individual measurement. The term systematic implies that the same magnitude Conditions in which random errors can result include: and sign of experimental uncertainty are obtained when 3 4 EXPERIMENT 1 / Experimental Uncertainty (Error) and Data Analysis the measurement is repeated several times. Determinate Avoiding systematic errors depends on the skill of the means that the magnitude and sign of the uncertainty can observer to recognize the sources of such errors and to be determined if the error is identifi ed. Conditions from prevent or correct them. which systematic errors can result include B. Accuracy and Precision 1. An improperly “zeroed” instrument, for example, an ammeter as shown in ● Fig. 1.1. Accuracy and precision are commonly used synonymously, 2. A faulty instrument, such as a thermometer that reads but in experimental measurements there is an important 101 °C when immersed in boiling water at standard distinction. The accuracy of a measurement signifi es how atmospheric pressure. This thermometer is faulty close it comes to the true (or accepted) value—that is, how because the reading should be 100 °C. nearly correct it is. 3. Personal error, such as using a wrong constant in cal- culation or always taking a high or low reading of a scale division. Reading a value from a measurement Example 1.1 Two independent measurement scale generally involves aligning a mark on the scale. results using the diameter d and circumference c of a The alignment—and hence the value of the reading— circle in the determination of the value of p are 3.140 can depend on the position of the eye (parallax). and 3.143. (Recall that p 5 c/d.) The second result is Examples of such personal systematic error are shown in ● Fig. 1.2. (a) Temperature measurement Figure 1.1 Systematic error. An improperly zeroed instrument gives rise to systematic error. In this case (b) Length measurement the ammeter, which has no current through it, would Figure 1.2 Personal error. Examples of personal error due systematically give an incorrect reading larger that the true to parallax in reading (a) a thermometer and (b) a meter value. (After correcting the error by zeroing the meter, stick. Readings may systematically be made either too which scale would you read when using the ammeter?) high or too low. EXPERIMENT 1 / Experimental Uncertainty (Error) and Data Analysis 5 (a) Good precision, but poor accuracy (b) Poor precision and poor accuracy (c) Good precision and good accuracy Figure 1.3 Accuracy and precision. The true value in this analogy is the bull’s eye. The degree of scattering is an indication of precision—the closer together a dart grouping, the greater the precision. A group (or symmetric grouping with an average) close to the true value represents accuracy. more accurate than the first because the true value of uncertainty of a number read from a measurement instru- p, to four figures, is 3.142. ment depends on the quality of the instrument and the fi neness of its measuring scale. When reading the value from a calibrated scale, only a certain number of fi gures Precision refers to the agreement among repeated or digits can properly be obtained or read. That is, only a measurements—that is, the “spread” of the measurements certain number of fi gures are signifi cant. This depends on or how close they are together. The more precise a group the least count of the instrument scale, which is the small- of measurements, the closer together they are. However, a est subdivision on the measurement scale. This is the unit large degree of precision does not necessarily imply accu- of the smallest reading that can be made without estimat- racy, as illustrated in ● Fig. 1.3. ing. For example, the least count of a meter stick is usually the millimeter (mm). We commonly say “the meter stick is calibrated in centimeters (numbered major divisions) with Example 1.2 Two independent experiments give two a millimeter least count.” (See ● Fig. 1.4.) sets of data with the expressed results and uncertain- The significant figures (sometimes called signifi- ties of 2.5 6 0.1 cm and 2.5 6 0.2 cm, respectively. cant digits) of a measured value include all the numbers The first result is more precise than the second that can be read directly from the instrument scale, plus because the spread in the first set of measurements one doubtful or estimated number—the fractional part of is between 2.4 and 2.6 cm, whereas the spread in the least count smallest division. For example, the length the second set of measurements is between 2.3 and of the rod in Fig. 1.4 (as measured from the zero end) is 2.7 cm. That is, the measurements of the first experi- 2.64 cm. The rod’s length is known to be between 2.6 cm ment are less uncertain than those of the second. and 2.7 cm. The estimated fraction is taken to be 4/10 of Obtaining greater accuracy for an experimental value depends in general on minimizing systematic errors. Obtaining greater precision for an experimental value depends on minimizing random errors. Rod C. Least Count and Significant Figures In general, there are exact numbers and measured numbers (or quantities). Factors such as the 100 used in calculating percentage and the 2 in 2pr are exact numbers.