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phy124on:error_and_uncertainty [2013/01/11 18:41] jhobbs |
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If you do **not** check the box, and, therefore, do **not** force the fit to go through the origin (0,0), the plotting program will find a value for the intercept $b$ and its uncertainty $\Delta b$, and they will be also be printed out in the upper-left corner of the plot. | If you do **not** check the box, and, therefore, do **not** force the fit to go through the origin (0,0), the plotting program will find a value for the intercept $b$ and its uncertainty $\Delta b$, and they will be also be printed out in the upper-left corner of the plot. | ||

+ | |||

+ | ==== The max-min method (be careful for "constrained" vs. "unconstrained" fits) ==== | ||

Another technique you can use to estimate the error in the slope is to draw "max" and "min" lines. Here we use | Another technique you can use to estimate the error in the slope is to draw "max" and "min" lines. Here we use | ||

- | our "eyeball + brain" judgment to draw two lines, one that has the maximum slope that seems reasonable, the "max" line, and another that has the smallest slope that seems reasonable, the "min" line. We do NOT use the computer to draw these lines, and normally we do the judgment process leading to our choice of suitable "max" and "min" lines on paper, but you can do it more quickly and simply by holding a clear plastic ruler up to the screen of the computer monitor to decide where you think the max and min lines should be. But please DON'T draw on the screen of the computer monitor! The surface exposed to you is made of soft plastic and can easily be scratched permanently. Such scratches distort the image being presented on the screen. A line is reasonable if it just passes within //most// of the error bars. You then just take two convenient points on the line, and find the change in the dependent variable "$y$" over the change in the independent variable "$x$" to calculate the slope. Doing this to work out the slope of both lines, max and min, gives you an estimate for the uncertainty in the slope. (Note that if you decide **not** to force your eyeball + brain max and min lines to go through the origin (0,0), you also get an estimate for the uncertainty in the $y$ intercept. This only makes sense if you did **not** "check the box" when using the plotting tool to do the linear fit.) | + | our "eyeball + brain" judgment to draw two lines, one that has the maximum slope that seems reasonable, the "max" line, and another that has the smallest slope that seems reasonable, the "min" line. We do NOT use the computer to draw these lines, and normally we do the judgment process leading to our choice of suitable "max" and "min" lines on paper, but you can do it more quickly and simply by holding a clear plastic ruler up to the screen of the computer monitor to decide where you think the max and min lines should be. But please DON'T draw on the screen of the computer monitor! The surface exposed to you is made of soft plastic and can easily be scratched permanently. Such scratches distort the image being presented on the screen. A line is reasonable if it just passes within //most// of the error bars. You then just take two convenient points on the line, and find the change in the dependent variable "$y$" over the change in the independent variable "$x$" to calculate the slope. Doing this to work out the slope of both lines, max and min, gives you an estimate for the uncertainty in the slope. |

+ | | ||

+ | === An example for the "unconstrained" case === | ||

+ | | ||

+ | Note that if you decide **not** to force your eyeball + brain max and min lines to go through the origin (0,0), you also get an estimate for the uncertainty in the $y$ intercept. This only makes sense if you did **not** "check the box" when using the plotting tool to do the linear fit. For a nice example of the max-min method being applied to a case where the linear fit is "unconstrained", viz., it is not forced to pass through the origin (0,0), see Fig. 4.1 in Sec. **4.3 Linear Relationships** on pp. 22-23 of | ||

+ | [[http://uregina.ca/~szymanss/uglabs/companion/Ch4_Graph_Anal.pdf|this document]] from the Department of Physics at the University of Regina. | ||

+ | | ||

+ | === Our example for the "constrained" case === | ||

The example we show next uses the same pendulum data presented above, but this time you should notice that the plot has been made "the other way", viz., as $L$ (cm) (on the $y$-axis) versus $T^2$ ($s^2$) (on the $x$-axis). You could do this yourself by entering the data into the plotting tool in the proper way. A consequence of plotting the data this way is that the large error bars -- those for $T^2$ -- are now in the horizontal direction, not in the vertical direction as they were for the first plot. This doesn't affect how we draw the "max" and "min" lines, however. You'll notice that the max and min lines for the present case, which appear in black on the computer screen versus green for the "best fit" line obtained with the plotting tool and versus red for the "error bars", both pass through the origin, as they should when one is comparing them to a constrained fit obtained by "checking the box", and they both pass through nearly all the "large" error bars, the horizontal ones. Your eyeball + brain choice of suitable max and min lines would undoubtedly be slightly different from those shown in the figure, but they should be relatively close to these. For the ones shown in the plot, which are reasonable choices, you may calculate yourself that the max line has a slope of about $\Delta y / \Delta x = 90/3.6 =25$ cm/s$^2$, and the min line has slope of about $\Delta y / \Delta x = 90/3.8 = 23.7$ cm/s$^2$. Therefore if you used this max-min method you would conclude that the value of the slope is 24.4 $\pm$ 0.7 cm/s$^2$, as compared to the computers estimate of 24.41 $\pm$ 0.16 cm/s$^2$. | The example we show next uses the same pendulum data presented above, but this time you should notice that the plot has been made "the other way", viz., as $L$ (cm) (on the $y$-axis) versus $T^2$ ($s^2$) (on the $x$-axis). You could do this yourself by entering the data into the plotting tool in the proper way. A consequence of plotting the data this way is that the large error bars -- those for $T^2$ -- are now in the horizontal direction, not in the vertical direction as they were for the first plot. This doesn't affect how we draw the "max" and "min" lines, however. You'll notice that the max and min lines for the present case, which appear in black on the computer screen versus green for the "best fit" line obtained with the plotting tool and versus red for the "error bars", both pass through the origin, as they should when one is comparing them to a constrained fit obtained by "checking the box", and they both pass through nearly all the "large" error bars, the horizontal ones. Your eyeball + brain choice of suitable max and min lines would undoubtedly be slightly different from those shown in the figure, but they should be relatively close to these. For the ones shown in the plot, which are reasonable choices, you may calculate yourself that the max line has a slope of about $\Delta y / \Delta x = 90/3.6 =25$ cm/s$^2$, and the min line has slope of about $\Delta y / \Delta x = 90/3.8 = 23.7$ cm/s$^2$. Therefore if you used this max-min method you would conclude that the value of the slope is 24.4 $\pm$ 0.7 cm/s$^2$, as compared to the computers estimate of 24.41 $\pm$ 0.16 cm/s$^2$. |