ERROR ANALYSIS ACTIVITY 1: STATISTICAL MEASUREMENT UNCERTAINTY AND ERROR BARS

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1 ERROR ANALYSIS ACTIVITY 1: STATISTICAL MEASUREMENT UNCERTAINTY AND ERROR BARS LEARNING GOALS At the end of this activity you will be able 1. to explain the merits of different ways to estimate the statistical uncertainty in common measurements like a DC voltage reading. 2. to explain which of the methods actually give an estimate of the standard deviation, which over/underestimate the uncertainty, and which are just a rough estimate of the uncertainty. 3. compute the uncertainty in a derived quantity using the uncertainties in the measured quantities (error propagation). 4. add error bars to a plot as a visual representation of uncertainty. INTRODUCTION TO MEASUREMENT UNCERTAINTY When attempting to establish the validity of our experimental results it is always important to quantify the uncertainty. Measurement uncertainty wasn t invented to make lab classes tedious, rather it is a core part of any experimental work that gives us a way to quantify how much we trust our results. A simple and rigorous way to make a measurement and estimate its uncertainty is to take N measurements {y 1, y 2,, y N } and estimate the value by the mean: N y = 1 N y i The estimated uncertainty (standard deviation, σ y, or variance σ y 2 ) of any one measurement is given by i=1 σ 2 y = 1 N N 1 (y i y ) 2 While the uncertainty in the mean value σ y is smaller and is given by i=1 σ y 2 = σ y 2 The remainder of this activity will discuss a variety of practical considerations about using uncertainty in the lab. N

2 ESTIMATING THE MEAN AND UNCERTAINTY The next four questions cover a basic measurement that was essential in the Gaussian Laser Beams lab. In a group of 2 or 3, find a lab bench with a laser and turn it on immediately. It might take 5 minutes for the laser to warm up. Use the photodetector to measure the DC optical power. Question 1 Measurement and uncertainty using the multimeter Note: It may be possible to answer questions 1 and 2 at the same time if you use a BNC T to send the voltage to the multimeter and oscilloscope at the same time. Make a table of estimated DC voltages from the laser and the corresponding uncertainties using the following methods a. Eyeball the mean. Eyeball the amplitude of the random fluctuations. b. If your multimeter has the capability, set the multimeter on max/min mode to record the V max and V min fluctuations over a certain time period. You can estimate the mean by (V max + V min )/2 and the uncertainty by (V max V min )/2. c. Record the instantaneous voltage reading on the multimeter N times and calculate the estimated uncertainty from the standard deviation. d. What is the resolution intrinsic to the FLUKE multimeter according to the spec sheet on the course website (under Hints ). (No measurement required) How does this compare to the observed uncertainty in parts a-c? Question 2 Measurement and uncertainty using the oscilloscope Continue the previous table of estimated DC voltages from the laser and the corresponding uncertainties using the following methods. For each method comment on if and how it depends on the setting for the time scale or voltage scale on the oscilloscope. a. Eyeball the mean. Eyeball the amplitude of the random fluctuations (no cursors or measurement tools). b. Use the measurement function on the scope to record the mean and RMS fluctuations. c. Use the cursors to measure the mean and size of fluctuations. d. Record the voltage from the oscilloscope N times and calculate the estimated uncertainty from the standard deviation. e. A comparison with the data sheet is difficult because so many factors affect the observed noise in the oscilloscope. You can find some information in the RIGOL oscilloscope datasheet on the course website (under Hints ). There is information about the resolution and the DC measurement accuracy. Question 3 Summary of Questions 1 and 2 (make sure to support your answers for each question): a. Did any methods overestimate the uncertainty? b. Did any methods underestimate the uncertainty? c. How reliable was eyeballing? d. Did the time scale or voltage scale affect any of the oscilloscope measurements? If yes, how? Does this tell you anything about how to use the scope? e. Suppose the light into the photodetector was not constant during the measurements (due to variations of the laser, room lights, etc.). In which estimates of the uncertainty will this be included? f. Which method(s) should give a true estimate for the uncertainty?

3 Question 4 Measurement using the NI USB-6009 DAQ a. Is the USB data acquisition device more like the multimeter, or more like the oscilloscope? (Just think back to your experience using the DAQ last time. No data taking for this part.) b. How would you go about estimating the uncertainty when using your DAQ? WRITING NUMBERS AND THEIR UNCERTAINTY The convention used in this course is that we 1) only display one significant digit of the uncertainty (two are allowed if the first significant digit is a 1) 2) display the measurement to the same digit as the uncertainty. The numbers 154±3, ±0.04, and 245.1±1.4 follow the convention. However, numbers copied from the computer are often displayed as machine precision with no regard for significant digits. Question 5 Mathematica generated the following fit parameters and corresponding uncertainties: a = ± b = ± How should the two Mathematica fit parameters above be rewritten? ERROR PROPAGATION: FROM MEASURED TO DERIVED QUANTITIES The quantity of interest in an experiment is often derived from other measured quantities. An example is estimating the resistance of a circuit element from measurements of current and voltage, using Ohm s law (R = V/I) to convert our measured quantities (voltage and current) into a derived quantity (resistance). Error propagation comes in when we want to estimate the uncertainty in the derived quantity based on the uncertainties in the measured quantities. Keeping things general, suppose we want to derive a quantity z from a set of measured quantities a, b, c, etc. The mathematical function which gives us z is z = z(a, b, c, ). In general, any fluctuation in the measured quantities a, b, c, will cause a fluctuation in z according to δz = ( z ) δa + ( z) δb + ( z a b c ) δc + (1) This equation comes straight from basic calculus. It s like the first term in a Taylor series. It s the linear approximation of z(a, b, c, ) near (a 0, b 0, c 0, ). However, we don t know the exact magnitude or sign of the fluctuations, rather we just can estimate the spread in δa, δb, δc, which we often use the standard deviations σ a, σ b, σ c. In this case, the propagated uncertainty in z is: σ z 2 = ( z a ) 2 σ a 2 + ( z b ) 2 σ b 2 + ( z c ) 2 σ c 2 + (2)

4 There are standard equations provided in courses like the introductory physics lab for the error in the sum, difference, product, quotient. These are all easily derived from the general formula. Question 6 Question 7 Question 8 Review things from long ago Apply the general error propagation formula in Eq. 2 to calculate the derived uncertainty σ z 2 in terms of the measurement uncertainties σ a and σ b when a. z = a + b b. z = a b c. z = ab In the case where a voltage V and current I are measured to derive the resistance R, use Eq. 2 to calculate the uncertainty in R in terms of the uncertainties σ I and σ V. A diffraction grating can be used to measure the wavelength of light using 2a sin θ = nλ where a is distance between the grating spacing, and θ is the angle of incidence and reflection, and n is the order of the diffraction peak. Note that n is an integer. If we derive λ from measurements of θ and a, use Eq. 2 to calculate the uncertainty in λ in terms of the uncertainty in a and θ: σ a and σ θ. ERROR PROPAGATION IN MATHEMATICA So far we have explored the use of Mathematica for basic data analysis and plotting, but Mathematica also has powerful symbolic math capabilities. One example where this can be helpful is for complicated error propagation calculations. The following bit of Mathematica code can calculate the propagated variance in the derived quantity symbolically. The code is available and explained in a Mathematica notebook Uncertainty_Propagation.nb on the course website. (There is a link on the Lecture Activities page.) Question 9 In the Gaussian Laser Beams lab, you modeled the Gaussian beam s width w(z) as: w(z) = w ( z z 2 0 πw 2 0 /λ ) For the output beam of one of the lasers, a fit of beam width versus position gave the following fit parameters: z 0 = 0.03 ± 0.04 m The wavelength is given by λ = ± 0.1 nm. w 0 = (1.90 ± 0.09) 10 6 m Use Mathematica to estimate the uncertainty in the derived width w(z) when z is a distance of ± m from the waist position. You will probably find it helpful to copy and modify the code given in the example Mathematica notebook, Uncertainty_Propagation.nb on the Lecture Activities part of the course website.

5 ERROR BARS The error bar is a graphical way to display the uncertainty in a measurement. In order to put error bars on a plot you must first estimate the error for each point. Anytime you include error bars in a plot you should explain how the uncertainty in each point was estimated (e.g., you eyeballed the uncertainty, or you sampled it N times and took the standard deviation of the mean, etc.) ERROR BARS IN MATHEMATICA Creating plots with error bars in Mathematica requires the use of the ErrorBarPlots package. Suppose you had estimated the uncertainty at every point in a width measurement of your Gaussian laser beam to be 0.04 V. This error was chosen to demonstrate the mechanics of making a plot with error bars, but the uncertainty in the actual data was probably smaller than this. Also, there is no reason to believe it was the same at low and high voltages. The data and uncertainty form a three column data set. Micrometer Position (inches) Photodetector Voltage (V) Estimated uncertainty (V) The Mathematica code for plotting with error bars will look something like this: Needs["ErrorBarPlots`"]; d = Import[ Gaussian_beam_data_with_errors.txt, TSV ]; dnew = Table[{ {d[[i,1]], d[[i,2]]}, ErrorBar[d[[i,3]]] },{i,2,length[d]}]; ErrorListPlot[dNew, FrameLabel->{"Position, x (inches)", "Photodetector output (V)"}]

6 The first line loads the ErrorBarPlots package. The second line is important because the error bars have to be input in a certain format. The following table gives a comparison of plotting data with and without error bars in Mathematica. Here we are only adding y errors, and assume the positive and negative errors are the same. No error bars d = {{x1,y1}, {x2, y2},...} (built in function) ListPlot[d] With error bars d = { {{x1, y1}, ErrorBar[e1]}, {{x2, y2}, ErrorBar[e2]},...} Needs["ErrorBarPlots`"]; ErrorListPlot[d] EXAMPLE: GAUSSIAN LASER BEAM WIDTH MEASUREMENT Question 10 Import the dataset gaussian_data_with_errors.txt with the uncertainties from the website (in the Lecture Activities tab). Use the ErrorBarPlots package to reproduce a plot like this. If you need advice on setting plot options, look at the final section of the Mathematica Activity 2. REFERENCES 1. Taylor, J. R. (1997). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (p. 327). University Science Books. This is the standard undergraduate text for measurement and uncertainty. 2. Bevington, P. R., & Robinson, K. D. (2003). Data Reduction and Error Analysis for the Physical Sciences Third Edition (3rd ed.). New York: McGraw-Hill. Great for advanced undergrad error analysis. Professional physicists use it too.