Plotting Graphs. Error Bars

Transcription

1 E Plotting Graphs Construct your graphs in Excel using the method outlined in the Graphing and Error Analysis lab (in the Phys 124/144/130 laboratory manual). Always choose the x-y scatter plot. Number each graph and provide a descriptive title so that it is clear what is being graphed. Avoid stating titles in the simple form Velocity Versus Time. Velocity as a Function of Time for a Falling Ball tells the reader more about the graph. Label each axis clearly, stating the variable and its units. You may wish to change the scale or range of your graph, and this can be done by right-clicking on one of the axes and selecting FORMAT AXIS. On AXIS OPTIONS, you can then set the minimum or maximum values for that axis. Sometimes you may want to zoom in as close as possible to the data points so that they fill the entire graphing area. At other times, you may need to expand the scale to show the y-intercept. Error Bars As discussed in Appendix D, all measurements have an associated error. This error range should be represented on your graph by error bars and will be used to calculate the uncertainty in the graphical values. By right-clicking on one of the points on your graph, you can choose ERROR BARS from the LAYOUT tab to allow you to add x- and y-error bars. You can choose a fixed error value for all the points, or you can select a column of numbers from your spreadsheet to have a different error for each point.

2 If, after you add the error bars, they are too small to show on the scale of your graph, you must make a clear note on the graph that the error bars are too small to show. If one of your values appears anomalous, then draw a question mark beside it on your printed graph. In order for Excel to ignore that point, we must make a new graph with that point removed. You can remove that point from the graph simply by excluding it when selecting the data you use to make the graph. Best-Fit, Maximum, and Minimum Lines As discussed above, linear relationships are the easiest to analyze. Once you have plotted the data so that it follows a linear relationship, Excel can draw the best-fit line for you and determine the best-fit slope and intercept. In order to determine the error in your slope and intercept values, maximum and minimum lines are drawn on the graph. The maximum line is a reasonable estimate of the line with the greatest slope that can be drawn through the data, considering both the scatter of the data about the best-fit line and the size of the error bars. The minimum line is a reasonable estimate of the line with the least slope that can be drawn through the data with the same considerations as the maximum line. Because the lines are placed by hand, their positions will vary to some extent depending on the person drawing them. Note: Students often feel that their minimum and maximum lines should pass through the two points that are the farthest away from each other. There is no reason for doing this. Look at your graph and use the following three general scenarios to help you draw your lines: a) points with significant error bars: Your minimum and maximum lines should pass through the error bars of almost every point. Make sure to consider both the x- and y-error bars. maximum? best-fit minimum 108

3 b) scattered points with insignificant error bars: Use the scatter of the points to draw the minimum and maximum lines. In this case, it is likely that you have underestimated the error since the scatter in the data is much larger than the size of the error bars. best-fit minimum maximum c) almost perfectly linear points with insignificant error bars: Do not draw minimum and maximum lines. Instead, take a 1% error in the slope, and the vertical height of a datum point as the uncertainty in the intercept. To use this method, you must have both insignificant error bars AND perfectly linear points. This will not be the case for most of the graphs you produce. 109

4 Calculating the Slope and Intercept with Errors You now need to determine the slope and intercept values for your best-fit, minimum and maximum lines. The minimum and maximum lines will allow us to determine the uncertainty in the best-fit slope and intercept. There are two ways you can determine these values: 1) draw the lines by hand and manually calculate their slopes and intercepts, or 2) use the Excel Max/Min add-in to add these lines to your graph electronically and let Excel determine the slopes and intercepts for you. By hand You must determine the slope and intercept of each of the best-fit, maximum and minimum lines. The slope of a line is calculated using two points on the line, (x 1,y 1 ) and (x 2,y 2 ), where slope = rise run = m = y 2 y 1 x 2 x 1 Once the slope of a line is known, the intercept of the line is calculated simply by rearranging the equation for a straight line, where intercept = b = y mx m is the slope of the line you determined above and (x, y) is a point on the line (say either (x 1,y 1 ) or (x 2,y 2 ) that you used to calculate the slope above). Alternatively, you can read the intercept off of the graph as the y-value where the line intersects the y-axis. You must use this process for the maximum and minimum lines, but you can let Excel handle the best-fit line. Use Excel to add a linear trendline and display its equation on the graph before you print it. You can add the maximum and minimum lines by hand after printing. Excel Max/Min Add-in Instead of drawing them by hand, you can use a custom Excel add-in to add maximum and minimum lines directly to your graph. First, create your graph. Make sure you do not include empty cells in your data range when creating your graph. Then, add x- and y-error bars, a linear trendline, and the graph and axes titles. You do not need to display the trendline equation on the chart. Once you are satisfied with the look of your graph, move it to its own sheet. When you have the graph selected, the CHART DESIGN tab will appear in the Excel ribbon at the top of the sheet. Controls for the Max/Min line add-in appear under this tab. Clicking DISPLAY IN- FORMATION produces a message box with instructions for using the add-in. Click ADD MAX/MIN LINES to have Excel draw maximum and minimum lines on your graph. Two lines appear with different colours and different line styles, and you can choose which one to use as your maximum line and which to use as your minimum line. The lines appear in default positions that are not likely to be the best positions for the lines. You must move the lines into reasonable positions for the maximum and minimum lines, as discussed above. 110

5 You can click and drag a line to a new position. With a line selected, you can also click each of its endpoints and drag them to new positions. This lets you change the slope of the line. Once you are satisfied with the positions of your maximum and minimum lines, click CALCULATE MAX/MIN LINES. Excel will add the equations for the best-fit, maximum and minimum lines onto the top left corner of the graph, displayed with colours matching the corresponding lines. Note that the equations are written in scientific notation, and will look something like y =1.234E+01x E 02 The slope m is the number before the x, which in this example is The intercept b is the second number, which in this example is The letter E is often used in computing to mean 10 to the power of. You can change the positions of the lines after calculating their equations, but remember to click CALCULATE MAX/MIN LINES again to update the equations displayed on the graph. Important Note: If you move the location of your graph after adding the maximum and minimum lines, Excel may not be able to find it when you then need to calculate the equations for those lines, and you will get an error message. To avoid this, it is best to calculate the equations for the maximum and minimum lines as soon as you are done positioning them. Pay attention to any error message that appears when using the add-in, as it will likely guide you as to what to do! Calculating Uncertainty in Slope and Intercept To determine the error in the intercept, calculate the absolute discrepancy between the maximum and best-fit values, and the absolute discrepancy between the best-fit and minimum values E 1 = b max b best E 2 = b best b min The larger of these two values is then the error in the intercept, b. Similarly, the larger of E 3 and E 4 represents the error in the slope, m, where E 3 = m max m best E 4 = m best m min Thus, you quote (making sure to use the correct number of significant figures) slope = m best ± m intercept = b best ± b Make sure to show all of the above calculations in your lab report. Either include them in the analysis section or write them directly on the graph (as long as there is ample space), as on the next page. 111

6 Graph 1 - Height versus Square of the Time for a Falling Object y = 4.878x max best min 2 Height (cm) Time Squared (s 2 ) Calculations for Graph 1 b max = 0.2 cm b best = cm b min = cm E 1 = 0.2 cm E 2 = 0.1 cm m max = = 5.52 cm/s E 3 = 0.6 cm/s 2 m best = cm/s 2 E =07cm/s m min = 4.21 cm/s 2 Therefore: b =0.0 ± 0.2cm and m =4.9 ± 0.7cm/s