Using Graphical Analysis 1. Find the program at Start/Programs/Vernier Software/Graphical Analysis 3.4. (It may be on the Desktop already.) 2. Setting up the Data Table Double click on the X cell at the top of the first column to bring up the Column Options dialog box. In the Column Definition tab, input the name with uncertainties (Name), the symbol (Short Name), and the units for the data (Units). You may need to resize the column heading for the full name to display. If you would like to insert a symbol or exponent, click on the down arrow button off to the side. Click on the Options tab and pull down the Displayed Precision menu to choose an appropriate number of decimal places (look at your data). Then include your error bars by clicking on Error Bar Calculations/ Fixed Value/Error Constant and putting the uncertainty value for X here. (Alternatively, if the uncertainties are different for each data point, you can put them all in a separate data column and then choose Error Bar Calculations/Fixed Value/Use Column and then use the pull- down menu to choose the appropriate uncertainty column.) Repeat this procedure for the Y column. But remember, you only need error bars on one of the axes – chose the most significant (proportionally largest) error bars. 3. Inputting Data Input your data for both the X and Y columns. Be sure that you have the correct number of decimal places on your data. If not, go back to step 2 and fix the Displayed Precision. Did you remember to think about whether or not to include the data point (0, 0) – is it appropriate? 4. Setting up the Graph From the top toolbar, choose Options/Graph Options to get the dialog box. (If this option is not available because it is “grayed out,” click on the graph window to make it active. Alternatively, you can double click on the graph window.) In the Title box, type an appropriate title for the graph. Uncheck Connect Points. Remember, we don’t play “connect the dots.” Be sure that X Error Bars and/or Y Error Bars are both checked if you are using uncertainties. Use Point Protectors if there are no error bars. Click on the Axes Options tab. Be sure the correct data is being graphed on the X and Y axes. (There is an option for a second Y axis. Ignore this for now.) Choose Scaling – Autoscale from 0 (unless you have skipped a large section of numbers on one or both axes – a rare event) for each axis. 5. Best-Fit Line and Specific Experimental Equation If you think the relationship is linear (if not, skip to step 7), then go to the top tool bar and choose Analyze/Linear Fit to automatically place the best-fit line on your graph. The best-fit line will be drawn and an information box will be displayed containing the slope and y-intercept of the best-fit line. Drag this box to a part of the graph window where it will not obscure the graph. Use these values to write the specific experimental equation of the relationship. Be sure to use the correct number of sig figs. 6. Max/Min Lines To have the program draw in your max and min lines, you need to choose Analyze/Curve Fit from the top tool bar. Choose a Linear relationship from among the General Equations and for Options: Fit Type: choose Automatic. Click on Try Fit at the bottom of the box. A best-fit line will be displayed on the small graph in the upper left corner. You should now alter that line to find the line with the maximum possible slope that will fit within your error bars. You alter it by clicking on the up/down arrows next to the Slope and Y-Intercept in the Coefficients section. Playing with both of these will both translate and rotate your line until you are satisfied. You can change the amount of increase or decrease for each click by clicking on the Delta box (small triangle in a box next to the up/down arrows) and typing in an appropriate increment. When you have the max line, click OK to put it on your main graph. Drag the info box to a place where it will not cover the graph. Then do this whole process over again to draw your min line. Try to have all three lines that are now on your graph cross near the midpoint, but don’t go crazy if they don’t. An easy way to draw max/min lines is by connecting the top(left) of the first error bar to the bottom(right) of the last error bar (for a minimum line) and the bottom(right) of the first error bar to the top(left) of the last error bar (for a maximum line) unless the first or last points are clear outliers – use your judgment. If you wish to remove a line and start again, click on the X box in the upper left corner of its info box. This process is a little tedious but the good news is that you now don’t have to calculate any slopes by hand – they’re all listed in the info boxes associated with each line. Use these values to find the uncertainty on the slope of your best-fit line. Be sure to use the correct number of sig figs. 7. Best-Fit Curve If the relationship is not linear, choose the best-fit curve (parabola, hyperbola, etc.) by going to Analyze/Curve Fit from the top tool bar. (Make sure you click on some empty space in the Graph Window to make it active first.) Choose from among the General Equations and click on Try Fit. See which curve seems to fit the data best. Choose OK for your final choice. (Don’t forget that a square-root relationship y = c√x can be expressed as y = cx1/2 = cx0.5 .) 8. Text Window If you are printing up this page to add to a lab rather than cutting and pasting it into a Word document, type your name, date, and the title of the lab, as well as any other relevant information here. 9. Print a) If you are adding this to a lab handout, resize the Data Table Window, Graph Window and Text Window until they fit on the page neatly. You can check this by selecting File/Print Preview. BIG TIP: Be sure to select the Landscape orientation from the File/Page Setup option so the information will fit on the page neatly. b) If you are incorporating this into a full lab write-up, select the graph and cut and paste it into your Word document in the appropriate location. Although the graph will cut and paste relatively easily, the data table will not. You can get around this by doing a Print Screen (hitting the keyboard key that says print scr ) and copying the screen shot into a program like Paint. Then you can cut and paste the data table into your Word document. It’s easy to do and you can see me for details if you’re having trouble doing this. 10. Straightening the Graph If you need to linearize the graph, you can perform a mathematical calculation (like squaring or taking the reciprocal) on a whole column of data by choosing Data/New Calculated Column from the top toolbar. Type in the Name of the new variable and its Units. Use the drop-down arrow for superscripts and subscripts. Suppose time was plotted on the x-axis and you wanted to square it. Type in Time Squared and units of s2. Then, with the cursor in the Equation box, use the pull-down menu of the Variables (Columns) to choose which column of data to transform and then either type in a mathematical function or use the pull-down menu of the Functions. Using our example, you would choose the Time column from the drop- down Variables (Columns) list and then type ^2. If you wanted to square root a column, you would choose sqrt from the drop-down Functions list. Choose the Options tab to fix the number of decimal places (or sig figs) and remove the error bars. Remember, you don’t need error bars on transformed data since we haven’t yet learned how to calculate them. Finally, click Done. Doing all this will add a whole new data column to your data set that you can then plot. You will see this column if you scroll to the right in your Data Set window. You can plot this new data column instead of the old (untransformed) data column by double-clicking on the graph and choosing the Axes Options tab of Graph Options. Use the pull-down list for the X-axis to choose what to plot on the x-axis and then check which data set you want on the y-axis from the Y-Axis Columns list. (Be sure to uncheck the other data sets you don’t want to graph.) Now that you have a linear graph, you can draw in the best-fit line and then the max and min lines, just the way it was described in steps 5 and 6, since you should still have the error bars on the untransformed data.