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Measurement, Uncertainty, and Uncertainty Propagation 205 Measurement, Uncertainty, and Uncertainty Propagation 205 Name _____________________ Date __________ Partners ________________ TA ________________ Section _______ ________________ Measurement, Uncertainty, and Uncertainty Propagation Objective: To understand the importance of reporting both a measurement and its uncertainty and to address how to properly treat uncertainties in the lab. Equipment: meter stick, 2-meter stick DISCUSSION Understanding nature requires measuring things, be it distance, time, acidity, or social status. However, measurements cannot be “exact”. Rather, all measurements have some uncertainty associated with them.1 Thus all measurements consist of two numbers: the value of the measured quantity and its uncertainty2. The uncertainty reflects the reliability of the measurement. The range of measurement uncertainties varies widely. Some quantities, such as the mass of the electron me = (9.1093897 ± 0.0000054) ×10-31 kg, are known to better than one part per million. Other quantities are only loosely bounded: there are 100 to 400 billion stars in the Milky Way. Note that we not talking about “human error”! We are not talking about mistakes! Rather, uncertainty is inherent in the instruments and methods that we use even when perfectly applied. The goddess Athena cannot not read a digital scale any better than you. Significant Figures The electron mass above has eight significant figures (or digits). However, the measured number of stars in the Milky Way has barely one significant figure, and it would be misleading to write it with more than one figure of precision. The number of significant figures reported should be consistent with the uncertainty of the measurement. In general, uncertainties are usually quoted with no more significant figures than the measured result; and the last significant figure of a result should match that of the uncertainty. For example, a measurement of the acceleration due to gravity on the surface of the Earth might be given as g = 9.7 ± 1.2 m/s2 or g = 9.9 ± 0.5 m/s2 but 1 Possible exceptions are counted quantities. “There are exactly 12 eggs in that carton.” 2 Sometimes this is also called the error of the measurement, but uncertainty is the modern preferred term. Vanderbilt University, Dept. of Physics & Astronomy Modified from: RealTime Physics, P. Laws, D. Sokoloff, R. Thornton General Physics Part A, Spring 2010 and University of VA Physics Labs: S. Thornton 206 Measurement, Uncertainty, and Uncertainty Propagation not as g = 9.7 ± 1.25 m/s2 or g = 9.92 ± 0.5 m/s2. In the last two cases, the last significant figure of the result and uncertainty do not match. When multiplying or dividing two numbers that have the same number of significant figures but differ in magnitude (e.g. 125 × 5.25), the final results should be quoted to the same number of significant figures (656). When adding and subtracting a two numbers, the lowest number of decimal places is used to specify the number of significant figures. (10.3 + 11.256 = 21.6) To minimize errors in calculations due to round off during intermediate calculations, you should generally keep at least one additional significant figure than is warranted by the uncertainties in each number. Work out the following examples in your lab write-up: 1. The length of the base of a large window is measured in two steps. The first section has a length of 1=1.22 m and the length of the second section is 2=0.7 m. What is the total length of the base of the window? 2. A student going to lunch walks a distance of x = 102 m in t = 88.645 s. What is the student's average speed? Types of uncertainties A systematic uncertainty occurs when all of the individual measurements of a quantity are biased by the same amount. These uncertainties can arise from the calibration of instruments or by experimental conditions such as slow reflexes on a stopwatch. Random uncertainties occur when the result of repeated measurements vary due to truly random processes. For example, random uncertainties occur due to small fluctuations in experimental conditions or due to variations in the stability of measurement equipment. These uncertainties can be estimated from the distribution of values in repeated measurements. Vanderbilt University, Dept. of Physics & Astronomy Modified from: RealTime Physics, P. Laws, D. Sokoloff, R. Thornton General Physics Part A, Spring 2013 and University of VA Physics Labs: S. Thornton Measurement, Uncertainty, and Uncertainty Propagation 207 Mistakes can be made in any experiment, either in making the measurements or in calculating the results. However, by definition, mistakes can also be avoided. Such blunders and major systematic errors can only be avoided by a thoughtful and careful approach to the experiment. Estimating uncertainty By eye or reason: Measurement uncertainty can often be reasonably estimated from properties of the measurement equipment. For example, using a meter stick (with marks every millimeter), a straight line can be easily measured to within half a millimeter. For an irregularly-edged object, the properties of its edges may limit the determination of its length several millimeters. Your reasoned judgment of the uncertainty is quite acceptable. By repeated observation: If a quantity x is measured repeatedly, then the average or mean value of the set of measurements is generally adopted as the "result". If the uncertainties are random, the uncertainty in the mean can be derived from the variation in the set of observations. Shortly, we will discuss how this is done. (Oddly enough, truly random uncertainties are the easiest to deal with.) Useful definitions Here we define some useful terms (with examples) and discuss how uncertainties are reported in the lab. Absolute uncertainty: This is the magnitude of the uncertainty assigned to a measured physical quantity. It has the same units as the measured quantity. Example 1. Suppose we need 330 ml of methanol to use as a solvent for a chemical dye in an experiment. We measure the volume using a 500 ml graduated cylinder that has markings every 25 ml. A reasonable estimate for the uncertainty in our Vanderbilt University, Dept. of Physics & Astronomy Modified from: RealTime Physics, P. Laws, D. Sokoloff, R. Thornton General Physics Part A, Spring 2013 and University of VA Physics Labs: S. Thornton 208 Measurement, Uncertainty, and Uncertainty Propagation measurements is ½ of the smallest division. Thus we assign an absolute uncertainty to our measurement of ∆V = ±12 ml. Hence, we state the volume of the solvent (before mixing) as V = 330±12 ml. Relative uncertainty: This is the ratio of the absolute uncertainty and the value of the measured quantity. It has no units, that is, it is dimensionless. It is also called the fractional uncertainty or, when appropriate, the percent uncertainty. Example 2. In the example above the fractional uncertainty is Vml12 0.036 3.6% (0.13) Vml330 Reducing random uncertainty by repeated observation By taking a large number of individual measurements, we can use statistics to reduce the random uncertainty of a quantity. For instance, suppose we want to determine the mass of a standard U.S. penny. We measure the mass of a single penny many times using a balance and interpolate between divisions by eye. The results of 17 measurements on the same penny are summarized in Table 1. Table 1. Data recorded measuring the mass of a US penny. mass (g) deviation (g) mass (g) deviation (g) 1 2.43 -0.088 10 2.46 -0.058 2 2.49 -0.028 11 2.52 0.002 3 2.49 -0.028 12 2.4 -0.118 4 2.58 0.062 13 2.58 0.062 5 2.52 0.002 14 2.61 0.092 6 2.55 0.032 15 2.49 -0.028 7 2.52 0.002 16 2.52 0.002 8 2.64 0.122 17 2.46 -0.058 9 2.55 0.032 The mean value ¯m of the measurements is defined to be 11N mmmmm i 12. 172.518 g (0.14) N i1 17 Vanderbilt University, Dept. of Physics & Astronomy Modified from: RealTime Physics, P. Laws, D. Sokoloff, R. Thornton General Physics Part A, Spring 2013 and University of VA Physics Labs: S. Thornton Measurement, Uncertainty, and Uncertainty Propagation 209 The deviation di of the ith measurement mi from the mean value ¯m is defined to be dmmii (0.15) Fig. 1 shows a histogram plot of the data on the mass of a US penny. Also on the graph is a plot of a smooth, bell-shaped curve that represents what the distribution of measured values would look like if we took many, many measurements. The result of a large set of repeated measurements subject only to random uncertainties will always approach a limiting distribution called the normal or Gaussian distribution. The larger the number of measurements, the closer the data will approach the normal distribution. This ideal curve has the mathematical form: 2 1mm N Number() m e 2m (0.16) m 2 where N is the total number of measurements. The normal distribution is symmetrical about ¯m. Figure 1. The Gaussian or normal distribution for the mass of a penny N=17, ¯m =2.518 g, ∆m=0.063 g. We now define the standard deviation ∆m as NNmm22 mm mgii0.063 (0.17) ii11N 116 Vanderbilt University, Dept. of Physics & Astronomy Modified from: RealTime Physics, P. Laws, D. Sokoloff, R. Thornton General Physics Part A, Spring 2013 and University of VA Physics Labs: S. Thornton 210 Measurement, Uncertainty, and Uncertainty Propagation For standard distributions, 68% of the time the result of an individual measurement would be within ± ∆m of the mean value ¯m. Thus, ∆m is the experimental uncertainty for an individual measurement of m.
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