Computer Data Analysis and Use of Excel

Transcription

1 Computer Data Analysis and Use o Excel I. Theory In this lab we will use Microsot EXCEL to do our calculations and error analysis. This program was written primarily or use by the business community, so it is very user riendly and easy to learn. It also has most o the unctions and tools necessary or serious scientiic calculation and or producing beautiul, customized charts and graphs. Each lab station has an IBM PC, connected over a local network to a laser printer and to a more powerul PC where all the programs and data iles are stored. When the computers are turned on, you will need to log on with the user name student and the password student in order to have access to this network. II. Experimental Procedure A. Running Microsot Excel. This program is stored on the hard disk o each computer workstation. (Below, when we say open, choose, and select, that generally means to click on something with the mouse.) To run Excel, use the let mouse button to double click on the Microsot Excel icon located on the desktop screen. B. Simple calculations. You should now have an empty spreadsheet, titled Sheet1, open in ront o you. Select cell B2, type your name and press Enter. Notice that the contents o the selected cell appear on a line near the top o the window, as well as in the spreadsheet. You will be calculating sin(θ) and cos(θ) or θ ranging rom 0.0 to 6.0 radians in increments o 0.1 radian. Details about how to do this ollow. In general, ormulas to be typed into Excel will be in bold as ollows: please type this into Excel. Select cell B5 and enter the number 0. Now enter the number 0.1 in cell B6. Highlight cells B5 and B6 by clicking on cell B5 and dragging the mouse over cell B6. Move the mouse arrow to the bottom right corner o cell B6 (there should be a small black box in this corner, over which the cursor becomes a simple cross: +); push and hold the let mouse button and drag the mouse down column B. Column B should now contain a series o increasing values. Make sure that the value contained in the last cell, B65, is 6.0. Click on cell B4, and enter theta (rad). This column contains values o the angle theta, in radians. Next click on cell C5. Here we want to calculate the value o sin(0), the sine o the theta value in cell B5. To do this, type: =sin(. Now click on cell B5, in the bar near the top o the screen you should see: =sin(b5. Close the parentheses, and press the Enter key. The number zero should appear in C5; this is the value o sin(0). Select cell C5. Then move the cursor to the lower right-hand corner o cell C5, until the cursor changes to a simple cross, as beore. Hold the mouse button down, and drag the ormula all the way down the column, to cell C65, and release the button. An entire column o values o Pcintro- 1

2 sin(θ) values will be calculated, almost instantly. Now, click on cells C5, C6, and C7, one ater the other, and watch the ormula appearing in the data bar near the top o the screen. Do you see what happened? As the ormula was copied down, the cell number or theta was incremented, so that each cell calculates a dierent value o sin(θ). This clever trick is what lets you do a lot o calculations ast with a spreadsheet. To calculate cos(, repeat the process you used to calculate sin(θ) except now use the ormula: =cos(b5) or example. C Graphing. In this section you will graph sine and cosine o theta vs. theta (theta goes on the x-axis). Details about how to do this ollow. You should have three columns o numbers: values o theta in B5-B65, values o sin(θ) in C5-C65, and values o cos(θ) in D5-D65. To graph these values, irst select the entire block o data by putting the cursor on cell B5, and, holding the button down, dragging down to D65. The data should appear as a highlighted block, in contrasting colors. Click on the ChartWizard button, the third icon rom the right on the toolbar across the top o the screen. Now, indicate where to put the chart, by clicking once on the spot where you want your graph. A ChartWizard dialog box should open up, indicating that you are on step one o ive. Step 1: Veriy that the range o values to be graphed is correct. In this case, Range = $B$5:$D$65. Click on the Next> button to continue on to step 2. Step 2: This allows you to choose the type o chart you wish to display. For now choose XY (Scatter) by selecting the appropriate box and then clicking the Next> button. Step 3: Here you choose the ormat o the chart you are going to display, select box 2, and click on the Next> button. Step 4: Leave all the selections unchanged and click on Next> to go straight on to step 5. Step 5: Here, you choose whether or not you wish to display a legend (in general, only use a legend i you are graphing more than one line per graph). You can add a title to your chart in the Chart title: box, and you can name each o the coordinate axes. When you are done, click on the Finish button to complete your chart. At any point during this process, you can repeat any step by using the <Back button to retreat to any earlier step in the sequence. To see this graph better, single click on any portion o it, then move the mouse arrow over any o the six small boxes around its borders where the mouse pointer becomes a double-headed arrow. Push and hold the let mouse button and move the mouse to either enlarge or reduce the graph s size along that edge. Here are some things to try now, in any order: 1. Try some other types o graphs. Repeat the process again, but make your own choices or steps 2 and 3. See how these changes aect the displayed chart. 2. Select one o the curves by double-clicking on the chart, then double-clicking on the curve. You will get a menu where you can change the type o points, or eliminate points and make a line instead. In general, we will be graphing our data points. Pcintro- 2

3 3. Try moving the chart to a dierent location. Push and hold the let mouse button while the arrow is on the chart, then move the chart around the page by moving the mouse. Let go o the mouse button and the chart drops where you leave it. Beore you print your graph, make sure everything its on one page to save paper. To save ink, highlight the gray area o your graph, then right click on it and choose Clear rom the popup menu. This makes the background o the graph white instead o gray. You should do this beore printing out any graph in the uture. To make sure your graph and data look the way you want, preview a printout by selecting Print Preview rom the File menu, or by clicking on the print preview icon on the tool bar (the magniying glass over a sheet o paper, ith rom the let, top row). Here you can zoom in and out to preview the document, as it will be printed. When done, click on the Close box at the top o the preview screen. Print the document by selecting Print rom the File menu, then selecting OK rom the Print screen. Or you can print by simply clicking once on the print icon on the toolbar (the printer icon, ourth rom the let on the top row). Using the toolbar icon will not prompt you again, it will print the document as is! Close this workbook by selecting Close rom the File menu, or by clicking on the box containing an X just above the toolbar. You will be asked i you wish to save changes in Book 1, select no. D. Using Excel to perorm data analysis. Today most data analysis is done using computers. This section is an introduction to error analysis using Excel. We will be using these techniques in most labs this semester, so take good notes. Please make graphs o the ollowing data, print them out and place them in your lab book, according to the accompanying instructions. In each case: (1) label each axis appropriately, including the units in parentheses; (2) put a title at the top o the graph; (3) when you are graphing data, arrange to have the data either on the same page as the graph or printing, or on a acing page in your lab book rom where you paste in your printout. Open a new workbook by selecting New rom the File menu. Under General, select the Workbook, click on OK. You should now have a blank spreadsheet opened on your computer. 1. Trendline Below is a table o voltage measured across a resistor, or various values o the current. You will plot the data, with current on the x-axis and voltage on the y-axis. Then you will draw the best-it line through the points and determine its slope. Details about how to do this ollow. Begin by selecting cell B2, and illing in the names o all lab partners. It is important to do this on all graphs to properly identiy your work when printing in the lab. Next copy the table below into your spreadsheet cells A4 through C10 (row 4 should contain the headings: Measurement Number, Current (Amps) and Voltage (Volts). Notice the dierence in what happens when you press Tab and when you press Enter. Pcintro- 3

4 Measurement Number Current (Amps) Voltage (Volts) Plot the data, by irst highlighting cells B5 through C10. Then, click on the ChartWizard button on the toolbar and position the graph on your spreadsheet. In step 1, veriy that the Range = $B$5:$C$10. In step 2, select the X-Y Scatter option and then go to step 3. In this example, we want to include the gridlines on our graph, so select option (3) or the graph ormat selection then click Next>. Do not make any changes in step 4. In step 5, give your graph a title (in Chart title) and label your axes (including units in parentheses), then select Finish. Enlarge your graph to get a better view o your data. The slope o this line gives us important physical inormation: the value o the resistor. To get the slope, draw in a best-it line. To do this we will use the Excel unction Trendline. Select the graph, then select the data points. Right-click on the data points and select Add Trendline rom the popup menu. Choose linear regression (a straight line), and under Options, choose Display equation on chart. The best-it line should appear on your chart, plotted over the data points. The law o physics describing the voltage across a resistor is Ohm s law, which is written V = I R (1) where V represents the voltage across the resistor, I represents the current through the resistor, and R is the resistance. Q1. From Ohm s law, what is the theoretical y-intercept? (Hint, compare Ohm s law with the equation or a straight line.) Q2. For the data points you just plotted, what is the value o the resistance? 2. Linest In order to get a measure o how good your data is, we will use Excel to calculate the error o your slope and y-intercept using its LINEST unction. Highlight the our cells B13 through C14. Type =linest(c5:c10,b5:b10,1,1) and press the Ctrl, Shit, and Enter buttons simultaneously. The our squares should contain the ollowing inormation: B C 13 The slope: The intercept: The error on the slope: Pcintro The error on the intercept: The standard orm or writing your result (you should remember this as you will use it in the uture) o the experimental value o R is:

5 R = 2.71 Ohms ± 0.20 Ohms Note: you can require the intercept o the line to be zero, by selecting this option in the trendline method or by entering =linest(c5:c10,b5:b10,0,1) or the linest method. Print out your data table and graph and paste them in your lab book. I you want to save your ile, click on ile, save as and navigate to the proper directory. I you are in section n, use the directory: C:\labs\ph122\secn\ Now open a new workbook. 3. Semi-log Plot The exponential unction occurs requently in physical phenomena. For example, the voltage across a capacitor as it discharges through a resistor takes the orm: V(t) = V o e -t/τ (2) where V o is the maximum value o the voltage and the time constant τ determines how rapidly it decays. There are two times that are o interest: (1) the time constant, τ. (This is the Greek letter tau, rhymes with ow.) When t = τ, the voltage decreases by a actor 1/e or V(τ) V o /3. (2) the hal-lie, T 1/2 = 0.693τ. When t = T 1/2, the voltage decreases by a actor 1/2 or V(T 1/2 ) = V o /2. An exponentially decaying unction looks simpler when its logarithm is plotted as a unction o time. Taking the natural log o both sides o the above equation yields: ln(v) = ln(v o e -t/τ ) = ln(v o ) + ln(e -t/τ ) = ln(v o ) t/τ Q3. For the equation o the natural log o voltage, ln(v), what is the theoretical slope and what is the theoretical y-intercept? (Hint: Comparing the above equation with the equation or a straight line.) Now, you will make two graphs using the data below: a linear graph and a log-linear (also called semi-log) graph. You will need the natural log o the voltages, so irst calculate those numbers. Then make a graph o V (y-axis) versus time (x-axis), and a separate graph o the natural log o V, ln(v) (y-axis) versus time (x-axis). Determine the slope o the semi-log graph. Enter the table below into your spreadsheet starting with cell B4 or time (sec). You should have lnv in cell D4. In cell D5 use the unction =ln(c5) and press Enter. Complete the table using the method explained in part B above. Create a graph o voltage versus time using the method outlined above. Measurement Time (sec) Voltage (V) ln V Number Pcintro- 5

6 Now, create a second graph o the natural log o voltage versus time. This time you can highlight the pertinent cells by irst highlighting cells B5 through B14. Then, while pressing the Ctrl button, highlight cells D5 through D14. Now continue as usual. Once you have your graph insert a best-it line using the Trendline method described above. You may save your data using the procedure outlined above i you choose. Q4. Compare the two graphs you have created. Which one most resembles a straight line? Q5. What is the slope o the ln(v) vs. time graph? Q6. Use the value o the slope to calculate the time constant and the hal-lie or this data set. 4. Statistical analysis. When we measure a physical quantity more than once, we can use Excel to calculate error on this value. We can also compare our measured values with an expected (or theoretical) value. Suppose that a number, say the ocal length o a lens, has been measured twice. The dierence between the two measurements gives an idea o the random error in the measurement. We will deine some terms, and show how to do the calculation, using EXCEL or by hand. The measured variable. Here it represents the ocal length. The average value o. (Pronounce it "e bar.") σ The standard deviation o the set. This represents the error on a single measurement o. (Pronounce it "sigma e.") σ σ The standard deviation o the mean. σ =, where N is the number o N measurements (N = 2, in our case). This is what we will reer to as the error on the measurement o. (Pronounce it "sigma e bar.") disc The "discrepancy," or dierence between the measured value and an expected ("theoretical") value: disc = theor. # o sigmas The discrepancy, expressed in terms o the standard deviation o the mean: # o sigmas = disc/ σ. Q The "quality o agreement." This is a qualitative statement about how likely it is that a value measured with the precision indicated by σ should disagree with the true value by a certain number o sigmas, due to random errors. Look up Q in the Pcintro- 6

7 table on page 4 o the appendix on errors in this manual. Let us suppose that the two measured values o the ocal length o a lens are 11.5 cm and 9.2 cm, and that the lens is expected a priori to have a ocal length o 10 cm. We will calculate,, σ, disc, # o sigmas, and Q. Open a new spreadsheet. In B3 enter 1. In C3 enter 2. In D3 enter -bar. In E3 enter sig-bar. In F3 enter -theor. In G3 enter disc. In H3 enter No. o sigmas. And in I3 enter Q. In B4 and C4 enter the two values o, 11.5 and 9.2. In D4 enter =average(b4:c4). In E4 enter =stdev(b4:c4)/sqrt(2). In F4 enter 10, the theoretical value o. In G4 enter =abs(d4-f4), this takes the absolute value o the dierence. In H4 enter =G4/E4. Then look up the quality o agreement in Table I o the appendix on Theory o Statistics and enter it in I4. Your results should look like this: 1 2 bar sig--bar -theor disc No. o sigmas Q Very Good You now have the numbers to state the result o the measurement o. The standard way to state this result is: = cm ± 1.15 cm. Agreement with expected value is very good (0.3 sigmas). Be sure you understand this procedure, as it will be used repeatedly in the labs to come. Sometimes you may not have the computer available to do these calculations, and you will then need to do them by hand. Calculating the average value is easy, and dividing by the square root o two is also easy. To calculate the standard deviation by hand, you can use the ollowing ormula: σ = N i= 1 ( x i x) N 1 2 two values, and ( 1 ) + ( 2 ) 2 You should be able to plug in values or 1, 2, and and divide by the square root o 2 to veriy that Excel calculated the correct value or σ. Pcintro- 7