Metrology is the science of measurement and involves what types of measurements are possible, standards, how to properly represent a number and how to represent the uncertainty in measurement. In 1993 the International Standards Organization (ISO) determined that the probabilistic approach is the best method for data analysis. These recommendations have been adopted by most of the standards organizations around the world. Uncertainty for situations of one measured quantity: You have been directed to get an estimate of your reaction time and report the best value with uncertainty. You will also create a spreadsheet showing your work for this lab. This laboratory exercise is not a formal experiment and therefore you will not turn in a Formal lab Report. The purpose of this exercise is to build laboratory skill. You will turn in only the necessary data and results. Determining the uncertainty: Uncertainty calculation depends on the type of measurement taken, the type of instrument used, and the nature of the measurand. The result, the best reported value with the uncertainty is the measurement. Probability Density Function: Probability density, p, is the probability that a variable exists between two values. The probability density function, p(x), is the probability density as a function of the variable x. The area under the graph is the probability that the reading is between the bounds of the area. The area under the entire pdf is always 1. Three Types of Readings: Measurand: Type A: Multiple Readings. When it is possible to take multiple readings of the same things, a better representation of the situation is possible. It does not matter whether the readings are digital or analog, the methods are the same. Type B: Single Reading. In many cases, it is not possible to make multiple readings. In these cases, there are separate standards for representing analog and digital readings. Digital: For digital readings, the reading is given on the device and the decision about how many digits to report is easy. Report what is displayed. The reading is a rounding of the actual measurement and this rounding affects the pdf that determines the uncertainty. Analog: For analog readings, a judgement call must be made off of the scale presented. The number and size of the scale gradations will, in part, determine the width of the pdf and therefore, the uncertainty. This term refers to the physical quantity that is the result of the measurement activity. Page 1 of 9
Elements of a Reported Measurement: There are three elements that must be reported. The best approximation of the measurand. The measurand is represented by the center line on the pdf. The standard uncertainty. The symbol for standard uncertainty is u and is represented on the pdf as the half width of the shaded area. The pdf that is being used and is determined by the type of measurement made. Single Digital Readings: For example, what is the uncertainty of the time as read from a stop watch or timer? Determine the width of the rectangle by considering the rounding to the last digit displayed. The most probable reading is between and. Illustrate this with the rectangular pdf. Calculate the uncertainty using this pdf and the equation: u = a 2 3. Single Analog Readings: What is the uncertainty of y based on the placement of your finger on the ruler? This could be due to parallax or limited divisions on the ruler. You must decide where you can say that the probability is zero that the reading is beyond a particular point. In this way you determine the edges of the triangle s base. The most probable reading is between and. Illustrate this with the triangular pdf. Calculate the uncertainty using this pdf and the equation: u = a.. 2 6 Multiple Readings with Scatter: What is the uncertainty of y based on an assumption of a random, Gaussian, distribution? Using your spreadsheet, calculate the experimental standard deviation of the mean using the equations: u = σ n N i=1 (x i x) 2 (N 1) and σ = Where u is the standard deviation of the mean, and is the experimental standard deviation. Page 2 of 9
Figure 1: Buffler and Allie. (2009). Introduction to measurement in the physics laboratory: A probabilistic approach. Science Education Groupon. University of York. p. 74. Page 3 of 9
An example of single digital reading: As an example, consider a reading from a cell phone stopwatch. For a single reading of 0:19.2 we would read 0 minutes and 19.2 seconds. The diagram would then look like the graph shown below. u = a 2 3 u = 0.1 2 3 = 0.02887 Uncertainty is generally reported to 2 significant figures giving: u=0.029. However, the uncertainty has significant figures that are in digits beyond the digits measured. The value of 19.2 has its last significant figure in the tenths place. The uncertainty cannot be reported beyond that. The first thing is to round the uncertainty to 1 significant figure. Doing so we get u = 0.03. Still the hundredths place is beyond the allowed tenths place. So in this case we report u = 0.1. Notice that the shaded area of the rectangle is not the entire areas. It is only 58%, which is the 1 equation. We call this the coverage probability. When you report the value, you report: the best approximation +/- uncertainty with a coverage probability of % 3 in the You would then report: t = (19.2 +/- 0.1) sec. with a coverage probability of 58% using a rectangular pdf. a = 19.25 19.15 or t = 19.2 sec with a standard uncertainty of 0.1 sec. (with a 58% coverage probability, using a rectangular pdf). 19.15 sec 19.20 sec 19.25 sec An example of single analog reading: As an example, consider a reading from a meterstick. For a single reading of 24.53 cm we would read 24 cm and a little past the 5 mm mark. It is clear that the reading was no lower than 24.51 and no greater than 24.55. The diagram would then look like the graph shown below. u = a 2 6 Page 4 of 9
u = 0.04 2 6 = 0.00816497 Uncertainty is generally reported to 2 significant figures giving: u=0.0082 Notice that the shaded area of the rectangle is not the entire areas. It is only 65%, which is the 1 equation. We call this the coverage probability. 6 in the When you report the value, you report: the best approximation +/- uncertainty You would then report: a = 24.55 24.51=0.04 x = (24.53 +/- 0.0082) cm. with a coverage probability of 65% using a triangular pdf. However, once again the uncertainty is smaller than the smallest measured digit. Therefore we report, instead, x = (24.53 +/- 0.01) cm. with a coverage probability of 65% using a triangular pdf. 24.51 cm 24.53 cm 24.55 cm Combination of uncertainty from more than one effect on a single type of reading: Many times, uncertainty results from more than one contribution. For example, if the manufacturer claims that the instrument has a rated accuracy of 1% how would this change what is reported? Let s go back to the cell phone stopwatch and see how that might impact what we report. u 1 = 0.1 2 3 = 0.02887 u 2 = 0.01(19.20) = 0.192 The combination is done by thinking about each uncertainty as one side of a right triangle. The combination is the hypoteneuse. Notice that most of the uncertainty is a result of the larger of the two when one is much larger. A good diagram to explain this is: u c u 2 u 1 u y = u 1 2 + u 2 2 u c = 0.02887 2 + 0.192 2 u c = 0.1942 Page 5 of 9
76.51 76.52 76.53 76.54 76.55 76.57 76.58 76.59 76.6 76.61 76.62 Engineering Physics 221: Lab Uncertainty After rounding we have u c = 0.19 sec. And with a reading only to the tenths, we report u = 0.2 sec. You would then report: t = (19.2 +/- 0.2) sec. using a rectangular pdf. Height (cm) 76.52 76.54 76.51 76.57 76.55 76.53 76.54 76.57 76.58 76.59 76.60 76.53 76.55 76.55 76.57 76.58 An example of multiple readings with scatter: Next, let s consider uncertainty, when it is possible to gather many measurements of the same thing. Many students have measured the height of a projectile fired straight up. The list of values is given below. Notice that all the measurements are given to the same number of digits past the decimal. As you can see, there are a variety of measurements for the same distance. This variety has to do with uncertainty. All physical measurements, and therefore all data, have some degree of uncertainty. This tells the reader how accurate the measurement is. What you want to report is the most likely measurement. One way to find this from a list of many measurements is to find the average. Around this average are a number of other likely measurements. These can be reported as a range on both sides of the average. Notice that the histogram shown is roughly similar in shape to the Gaussian pdf. The following method assumes that your readings fall into a Gaussian shape. It is best to always check. 6 5 Projectile Height Frequency 4 3 2 1 0 See the spreadsheet example of calculations on the next page: Page 6 of 9
σ = s(d) = N i=1 (x i x ) 2 N 1 u = σ N Height (cm) Subtract each value from the average Square the differences 0.003636 1.32231E-05 76.52-0.03636 0.001322314 76.54-0.01636 0.000267769 0.003636 1.32231E-05 76.51-0.04636 0.002149587 76.57 0.013636 0.00018595 76.55-0.00636 4.04959E-05 0.003636 1.32231E-05 76.53-0.02636 0.000695041 0.003636 1.32231E-05 76.54-0.01636 0.000267769 76.57 0.013636 0.00018595 76.58 0.023636 0.000558678 0.003636 1.32231E-05 76.59 0.033636 0.001131405 76.6 0.043636 0.001904132 76.53-0.02636 0.000695041 76.55-0.00636 4.04959E-05 76.55-0.00636 4.04959E-05 76.57 0.013636 0.00018595 76.58 0.023636 0.000558678 0.003636 1.32231E-05 The average value Sum the squares of the differences 76.55636 0.01031 σ = s(d) = N i=1 (x i x ) 2 N 1 σ = s(d) = 0.01031 22 1 = 0.022157 u = σ N u = 0.022157 22 = 0.004724 Rounding to 2 sig. figs. u = 0.0047 You would then report: x = ( +/- 0.0047) cm. However, the digits do not match. So round to 1 sig fig and check. x = ( +/- 0.005) cm. Once again, u is too small for the digits allowed. So we finally report: x = ( +/- 0.01) cm. with a coverage probability of 68% using a Gaussian pdf. Page 7 of 9
Performing the Lab: Part I: Single readings When reporting always specify three items: The best approximation of the measurand (given by the location of the center of the pdf) The standard uncertainty u (calculated from the width of the pdf) The shape of the pdf you used. Single Digital Readings: For each case, calculate the uncertainty. Report the value with uncertainty. Include the rectangular pdf with center and width labeled. Show your calculation for uncertainty. Take one digital reading of the fastest start-stop time you can measure with a digital stopwatch or timer. Measure your reaction time using the tool at the following website: http://www.humanbenchmark.com/tests/reactiontime Remember that we are doing a Single Reading so only do this once. Can you figure out a way to use the photogates to measure your reaction time? Figure out a way and carry it out. Remember this is a single reading only. Single Analog Readings: For each case, report the value with uncertainty. Include the correct pdf with center and width labeled. Show your calculation for uncertainty. Use a meterstick to measure the size of a length of a block you will find in the lab. Calculate the uncertainty. Use a micrometer to measure the same length. Use a Vernier caliper to measure the same length. Part II: Multiple Readings with Scatter: One technique to measure reaction time is to calculate the time from the distance that a meterstick falls during the reaction time delay. Have one lab partner hold a meterstick so that a second partner s fingers are right at the top. The meterstick is then dropped without a warning. The second partner then closes their fingers to catch the meterstick. The location of the fingers with give the distance the meterstick has fallen. Record this distance as y. Repeat the meterstick fall measurement at least 20 times. Create a histogram of the data to determine whether it is Gaussian. Calculate the best possible value for the distance with its uncertainty. Report in the proper way for the type of measurement. Calculate your reaction time using y = 1 2 gt2, where g = 9.81 m s 2 In this case, t is a function of d. Therefore the uncertainty of t depends on the uncertainty of d. You have the uncertainty in d from part I. To find an expression for the uncertainty for time, we use differentiation. Page 8 of 9
Uncertainty of a function starts with: δt = dt δy. Here t is a function of y. Rearrange the equation dy below to solve for t. Calculate the derivative of t with respect to y, dt dy. Plug back into the equation δt = dt dy δy d = 1 2 gt2 t = dt dy = δt = ( )δy Rearrange to get: u t = 1 u y. Where the relative uncertainty in t is: u t. It is often easier to report t 2 y t uncertainty in the relative form as a percentage. You should now be able to calculate the uncertainty in time, u t. Report the best approximation of your reaction time with the uncertainty using the format described earlier. Compare your reaction time and uncertainty to the reaction time and uncertainty from the single measure found in part I. Make sure you have completed all the sections: Part I: Show calculations and results for all measurements. Part II: Show calculations for reaction time from multiple measures. Answers to questions: Compare the uncertainties for different measurement instruments in the single measurement types Compare reaction time between the single and multiple measure techniques. Page 9 of 9