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Unformatted text preview: Data Acquisition And Analysis Objective: To gain familiarity with some of the measurement tools you will use in lab this semester. To learn how to measure distance with a motion sensor and force with a force sensor. To acquire experimental data, present it both statistically and graphically and become skilled at its analysis and interpretation. Apparatus: Motion sensor, force sensor, LabPro interface, computer & keyboard Introduction This lab serves as an introduction to the tools you will use to gather and analyze experimental data. Data analysis is important in any scientific, technical or health-care discipline, where it is used for critical decisions and thinking. In addition, the familiarity you develop with these instruments and software will come in handy when they are used many times throughout the semester. In the first part of the lab you will generate data by measuring your own reaction time; your lab partner will then repeat the experiment and you will analyze the data and compare your results. In the second part of the lab, you will tinker with a motion sensor and a force sensor to become familiar with them. Theory of Uncertainty and Measurement A scientist or engineer collects experimental data by taking measurements. You have certainly collected experimental data in your life perhaps by finding out how much you weigh, how tall you are, or how fast you have run a certain distance. Let's assume that you know your height to be 5'6" (168 cm), your weight 140 lb (64 kg) and your time in the 100 meter dash 12.16 seconds how valid are these numbers? Are you exactly 168 cm tall, and not 168.0625 cm (5'6 1/8 ") tall? How does your 140 lb weight as measured on your $20 bathroom scale compare to your weight measured on a calibrated pharmaceutical scale used to weigh the ingredients at a manufacturing plant? To answer these questions, you will need to understand three basic concepts uncertainty, precision and accuracy. You may have used the last two interchangeably in everyday life, but here in the laboratory we will make an important distinction between the two. Precision and accuracy are two different things - you can have data with high precision but low accuracy, and vice versa. Uncertainty: There are two types of uncertainties you will encounter in the lab: Systematic Uncertainties and Random Uncertainties. Systematic uncertainties cause a measurement to be skewed in a certain direction, i.e., consistently large or consistently small. For example, weighing yourself repeatedly on a bathroom scale that has has an initial reading of 20 lbs will result in your always appearing to be heavier than you actually are. This form of uncertainty can be removed, once identified in this case you can just zero the scale using the little ridged knob (calibration). Random uncertainties are variations in measurement that linger even after systematic uncertainties have been eliminated. If you weighed yourself on ten bathroom scales around your neighborhood (or weighed yourself ten times on your bathroom scale) over the period of a few minutes, you would probably record ten different values even though you know your weight is essentially not changing. This sort of uncertainty cannot be eliminated but can be reduced by making lots of measurements and averaging. Precision: Measurement A is more precise than Measurement B if the uncertainty in A is less than that of B. If you weighed yourself 10 times on a bathroom scale, the difference between your highest and lowest readings may be as much as 5 pounds. However, at the doctor's office, the same measurement may only yield a difference of perhaps half a pound; we say that the latter measurement is more precise. Precision is affected by random uncertainty. Accuracy: Measurement A is more accurate than Measurement B if the result you get from A is closer to the generally accepted, or standard value, than B is. When you zeroed your bathroom scale, you improved the accuracy of the measurement. Accuracy is affected by systematic uncertainty. Note that you can have a high-precision, low-accuracy measurement, such as a doctor's scale that hasn't been calibrated. Conversely, you can have a low-precision, high accuracy scale, such as a cheap bathroom scale that happens to yield an average weight close to your true weight. Below is a representation of the permutations of accuracy and precision: Can you discuss the level of both accuracy and precision for each of the above dart games? MATHEMATICAL DESCRIPTION OF RANDOM UNCERTAINTIES The first concept is the average (arithmetic mean) of a number of measurements: x= i=1 N
N xi N (1) This formula says: Add the N measurements x1 , x2 , x3 , etc., up to xN . This sum is written as x i . Now divide by N to get the mean value of x. Another important statistical parameter is the
standard deviation, , which is a measure of the spread of the individual data points. A large value for implies a large spread in the values whereas a smaller value implies a small or narrow spread. We call (calculated from a set of measurements as shown below) the standard deviation for repeated individual measurements, x i . The standard deviation of the mean x of measurements gives the spread in the means of repeated sets of measurements x 1 , x 2 , x 3 ,... , x N . Mathematically, these are different:
i =1 = x i -x 2
1 N N -1 (2a) = xi -x 2
1 N [ N N -1] (2b) (Some definitions use N for , instead of N-1 which is more accurate.) A bin plot (histogram) of the number ni of individual measurement differences from the mean (in units of the standard deviation ), has a "bell shape "normal" distribution. The area under the normal curve centered around the mean between (x + ) and x ) is 68.3% of the total area under the curve. This means that about two-thirds of the measurements, xi, fall between (x + ) and (x - ). The area between (x + 2 )and (x - 2 ) is 95.4% of the total area. And, 3 covers 99.7%. The standard deviation is used as a measure of the random uncertainty expected for an individual measurement. We write 15.5 0.2 m for the mean and spread of individual measurements. This implies that the probability is about 66% that the true value lies between 15.3 and 15.7. However, in most science and engineering applications, you will need to calculate x , the mean and the standard deviation of the mean. To illustrate the procedure we will work out the mean value x and the standard deviation of a set of 21 individual data points, and then the predicted uncertainty, , of the set's mean. What does all this mean? In Psychology, someone with an IQ score within 1 of the mean is said to have average intelligence. Those scoring above +1 are considered to have above-average intelligence. Anything over +2 is termed gifted. For example, if the average IQ is 100, and the standard deviation is 15, then the average category consists of people scoring between 85-115 (68.3% of the population). Those scoring above 115 ( +1 ) comprise 2.3% of the population notice that this is half of 100% - 95.4% = 4.6% because we are only considering the part of the population with above-average intelligence. CALCULATION OF MEAN AND STANDARD DEVIATION USING INDIVIDUAL LENGTH MEASUREMENTS x i- x xi 15.68 0.15 15.42 -0.11 x i- x2 0.0225 0.0121 15.03 15.66 15.17 15.89 15.35 15.81 15.62 15.39 15.21 15.78 15.46 15.12 15.93 15.23 15.62 15.88 15.95 15.37 15.51 326.08 m -0.50 0.13 -0.36 0.36 -0.18 0.28 0.09 -0.14 -0.32 0.25 -0.07 -0.41 0.40 -0.30 0.09 0.35 0.42 -0.16 -0.02 (-0.05 m) 0.2500 0.0169 0.1296 0.1296 0.0324 0.0784 0.0081 0.0196 0.1024 0.0625 0.0049 0.1681 0.1600 0.0900 0.0081 0.1225 0.1764 0.0256 0.0004 1.6201 m2 From the above table we can make the following calculations for N = 21 measurements x i 326.08 x= i=1 = =15.53 m
N 21 N = = x i-x i=1 N 2 N -1
N = 1.6201 =0.29 m 20 x i-x i=1 2 [ N N -1] = 1.6201 =0.062m  Hence = 15.53m 0.062 m. This says the average is 15.53m, which average has an x uncertainty of 0.062m. But, the uncertainty or spread in individual measurements is = 0.29m. Remember, when you calculate a non-zero value for or random uncertainty is present; the values of tell you how large the magnitude. Increasing the number of individual measurements reduces the statistical uncertainty (random uncertainties); this improves the "precision". On the other hand, more measurements do not diminish systematic uncertainty in the mean because these are always in the same direction; the "accuracy" of the experiment is limited by systematic uncertainties. Often we must compare different measurements. Consider two measurements
A A B B If you want to compare, say, their difference, the expected uncertainty is given by 2 = 2 2 A B
2 The reason we sum the squares is that we must sum the squares of the variances x i- x , in order to get the standard deviation. (Look again at the above equations for the standard deviation.) We should expect that | A-B | < , if the two data sets are statistically the same. If they diverge greatly from the expected standard deviation, | A-B | > , they are then statistically different. Often, a value of reference for being significantly different. Measurement and Limitations 2 is used as a simplified Stopwatch.exe is software written in-house that measures your reaction time, specifically, the time interval between a visual/audio cue from your lab PC's monitor/speakers and your hitting your keyboard's space bar. A motion sensor is a device that continuously emits sound pulses which bounce off an object and detects them when they return to the sensor. By timing the time interval between the departure and arrival of the pulse (and knowing the speed of sound), the distance to the object can be determined continuously. The motion sensor you will use can measure objects between the range of 0.15 to 8 meters, and has a resolution (discussed later) of 0.001m (one millimeter). A force sensor is a device that continuously monitors the force exerted on it. There is a metal strip inside the sensor that deforms slightly in response to the force (stress) on it. Since the metal's electrical resistance varies as its dimensions change (strain), the electrical signal can be converted to a value for force this is basically a strain gauge. The force sensor you will use has a range of 10N and a resolution of 0.01N in low range and a range of 50N and a resolution of 0.05N in high range. You may remember from high school that a 1kg mass weighs 9.8N (2.2 lbs) at the surface of the Earth. Procedure (Reaction Time) For reaction time data, each partner should determine and compare her/his reaction time distributions and their means for the Space bar key on the keyboard. 1. Open Stopwatch.exe, a program in the Lab Softwares folder on the PC's desktop. Click on Reaction time; Click to start, leaving the pointer on the start box . Each time the taunting figure appears on the computer screen (with accompanying "whoop" sound, if speaker is on) click the Space bar to stop . The time in seconds between the signal and your response will be recorded. 2. Practice; select Stopwatch to stop, then Clear record. Start Reaction Time and collect data for five minutes. When you are finished with your last button press, Click on 'Stop watch' in the program so that it will no longer run the timer. 3. Select reaction times you have just generated by clicking and dragging with the mouse (or rightclicking with the mouse and choosing Select All), until you have gotten all the points. Once they have been highlighted, copy them to the Windows clipboard (first part of "copy and paste") either by using Ctrl-C (or right-clicking and choosing Copy). 4. Open Graphical Analysis, also in the Lab Softwares folder. You should see a gray-outlined data table on the left, and an empty graph on the right. You should see two data columns: X and Y. Double-click on the X heading, which brings up a windows called "Column Options". Type in "Trial" for Name and check the "Generate Column" box; click OK. Double-click on the "Y" heading and change the name to indicate which lab partner did the trial (for example, "Lisa"). Click on the first cell of this column and paste (Ctrl-V or right-click Paste) the data you copied from Step 3. 5. Data is automatically plotted in the graph window (Lisa vs. Trial). If you have any outliers (points which are obviously bad, such as those resulting from not being ready to click the button), remove them from the data column. Go to Analyze-->Statistics to determine the Minimum, Maximum, Mean and Standard Deviation. Note that this Standard Deviation is and not , but that you can get the latter from the former with minimal calculation by carefully examining the mathematical definitions for both. Record these values in the hand-in sheet. 6. Create a histogram, which is a plot of 'Frequency of Value' vs. 'Value'. Click on the existing graph window (so that the new histogram window you are about to create will have the same width) and select Insert-->Additional graphs-->Histogram. Double-click on the middle of the new blank graph and check the quantity (for example, Lisa) you wish to plot. Ideally you should see a bell curve shape if your histogram looks too "blocky", decrease the Bin Size (the resolution of the histogram) by double clicking on the histogram itself, then selecting the Bin and Count Options tab and entering a lower value in the Bin Size box. If you see bins with zeros in them, enter a higher value of Bin Size. 7. To check that your reaction time data's distribution indeed resembles a Gaussian, or bell curve, you will need to generate a "point plot" since the curve fitting function won't work on a histogram. Select Insert-->Graph, which will bring up a new blank graph window. Double click on the vertical (X) axis label of the graph, which will show you (via check boxes) the quantities to plot. Check "Lisa-Hist", which you can see after you expand (click on "+") on "Histogram" under "Y-axis Traces". Repeat for the horizontal (X) axis, this time selecting "Lisa-Bin". You should now see points rather than bars on your graph. 8. Normal curve-fitting procedure is to go to Analyze-->Curve Fit-->select Gaussian in General Equation and click on Try Fit. However, this will most likely yield a straight line, so you may have to help the fitting program as follows: a) Increase bin size if hist-data is too widely scattered. b) Manual fit, to provide reasonable starting parameters for the auto fit. Input estimated A (maximum height), B (center of peak) and C (width), D (vertical offset). Adjust these parameters for visual fit. Then switch to automatic and Try It. If the D parameter remains, the auto fit can sometimes do really stupid things, such as finding a large positive A and a large negative D, with the difference being around your manual fit starting value for A, or giving a horizontal line fit. If so, try getting rid of the D parameter, then c) Define function - choose Gaussian, then remove D parameter (vertical offset), manual fit, switch to auto and Try It. 9. The other lab partner should repeat Steps 1-8. Print out only one curve-fitted point plot from either partner, making sure it is properly labeled (Name, Lab Partner's name, Section, Date). Procedure (Motion and Force Sensor) Motion Sensor 1. Open "Motion PRACTICE.CMBL", which is a Logger Pro file in the same folder as this write-up (PC Desktop-->Course Folders-->205-->Measuring Motion). If you get a message warning you that there is data stored in the LabPro interface, choose the option to Ignore/Erase the data. You should see a window (Distance) on the left and a data Table Window on the right. Once you start collecting data, the table on the right will fill with Distance data. The plot on the left will also be drawn. Note that the time scale for graph ranges from 0 to 10 seconds the software will take data for 10 seconds at a time. 2. Position the motion sensor near the side of the table as close as possible to the edge, but far enough such that it still lands on the table in the event it tips over. Rotate the head so that the grilled circle points horizontally. 3. Play around with the apparatus and software. Still in Motion PRACTICE.CMBL, Press the Collect button in Logger Pro to collect data. You will hear rapid clicking, which is the sound of the pulses coming out of the motion sensor. Have your partner move back and forth in front of the motion sensor in a random fashion. 4. After the 10 second collection period, examine your graphs. In the menu bar, go to Analyze-- >Examine. Now activate any of the three graphs by clicking on it, then moving the cursor around to trace the plot you should see the coordinates displayed on the top left hand corner of the graph, and they will change as you move around the plot. If you are not getting any data, make sure your motion sensor is plugged into the DIG/SONIC1 input on the LabPro (translucent aqua blue covered flat box). 5. Mark off two lines on the floor corresponding to two positions along the beam line that you will position yourself on. Using both the motion sensor and a meter stick (separately), measure the distance between these lines. Do they agree? Force Sensor 1. Close down Motion PRACTICE.CMBL. Open "Force PRACTICE.CMBL", which is a Logger Pro file in the same folder as this write-up (PC Desktop-->Course Folders-->205-->Measuring Motion). 2. Press the Collect button, as you did with the motion sensor, gently pull or press on the hook at the end of the sensor and watch the graph on the screen. Which way is the force positive? Which way is it negative? Hold the sensor so that the hook is down and collect data is the reading higher than when the hook is up? If so, why? What about when the sensor is laying on the table how does this reading compare to hook-up and hook down? Again, explain. Assessment and Presentation (Hand-in Sheet/Lab Notebook) Fill in the tables below in your hand-in sheet and answer the questions there: Partner 1 Times x x MIN x MAX Partner 2 Times x x MIN x MAX Calculate the difference in the reaction times of you and your lab partner. Estimate the uncertainty in the difference. (HINT: You can determine from if you can find a simple relationship between their mathematical definitions). Questions 1. Were your and your lab partner's reactions times statistically similar or different? Explain. What criteria did you use? 2. What if your hand was placed on your head, instead of being held right over the keyboard's space bar? Would that affect your random uncertainty? Your systematic uncertainty? 3. What applications (real life uses) can you think of for a motion sensor? 4. What applications can you think of for a force sensor? ...
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