Error In Slope
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Springs Charge of an Electron Video Analysis Torque Rolling Objects Mechanical Energy Human Performance Old labs Extra Stuff Slope uncertainty Reports How to find the uncertainty in the slope This is an issue that I have not really addressed much. However, it is important enough that I talk about it. In many labs, you will collect data, make a graph, percent error slope find the slope of a function that fits that data and use it for something. Well, what if
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need to find the uncertainty in the slope? How do you do that? There are a couple of ways you can do this, neither are absolutely correct. However,
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if you write a formal lab report and you find the slope you MUST find the uncertainty in it. Here is some sample data. Suppose I measure the diameter and the circumference of several roundish objects. Here is my data. So, I want to plot
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this and find a functional relationship between these two. With error bars, this is what it should look like: Now, I want to fit a linear function to this data. That should be ok, but what about the uncertainty? Method 1 - use uncertainty of data points I could get the ratio of C/d by just looking at each data point. This is not as good as the slope because the slope essentially uses all the data points at once. In this method, I am going to find the slope as error in slope linear regression normal. In Excel, you could fit a trendline. Or, you could draw a best fit line. Either way, I would get something like this (I did this in Logger Pro): This gives a slope of 3.28 (compare to pi = 3.14). I could get a better slope if I required the fitting function to go through the origin (0,0), but I am not going to do that. In essence, the slope is: But, what if I just use one set of data points? Then I could use propagation of error as usual. This would give Where the delta - slope represents the uncertainty in the slope. For this method, just pick the data pair with the largest uncertainty (to be safe) - although hopefully, it won’t matter much. For this case, I will pick d= 0.06+/-0.002 m and C = 0.183 +/- 0.004 m. This would give an uncertainty in the slope of 0.2. I would write: (there are no units - they canceled) Method 2 The next method is better but a little tricky. Basically, if you draw a best fit line you could “wiggle” the line and still have it fit. This would give a minimum and a maximum slope. If you do the trendline in Excel, I am not sure how you would do this. If you do it by hand on graph paper, it would be easy. Here, I am going to manually fit two lines in Logger Pro. Here is the max slope that ‘looks’ like it fits. This gives a min slope of 3.0 and a max of 3.2 for an uncertainty of +/- 0.1. Announc
the picture below, the data points are shown by small, filled, black circles; each datum has error error in slope excel bars to indicate the uncertainty in each measurement. It appears that error in slope of linear fit current is measured to +/- 2.5 milliamps, and voltage to about +/- 0.1 volts. The hollow triangles slope uncertainty calculation represent points used to calculate slopes. Notice how I picked points near the ends of the lines to calculate the slopes! Draw the "best" line through all the points, https://www2.southeastern.edu/Academics/Faculty/rallain/plab193/page1/page35/page36/page36.html taking into account the error bars. Measure the slope of this line. Draw the "min" line -- the one with as small a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line. Draw the "max" line -- the one with http://spiff.rit.edu/classes/phys369/workshops/w2c/slope_uncert.html as large a slope as you think reasonable (taking into account error bars), while still doing a fair job of representing all the data. Measure the slope of this line. Calculate the uncertainty in the slope as one-half of the difference between max and min slopes. In the example above, I find 147 mA - 107 mA mA "best" slope = ------------------ = 7.27 ---- 10 V - 4.5 V V 145 mA - 115 mA mA "min" slope = ------------------ = 5.45 ---- 10.5 V - 5.0 V V 152 mA - 106 mA mA "max" slope = ------------------ = 9.20 ---- 10 V - 5.0 V V mA Uncertainty in slope is 0.5 * (9.20 - 5.45) = 1.875 ---- V There are at most two significant digits in the slope, based on the uncertainty. So, I would say the graph shows mA slope = 7.3 +/- 1.9 ---- V Last modified 7/31/2007 by MWR. Copyright © Michael Richmond. This work is licensed under a Creative Commons License.
treated statistically in terms of the mean and standard deviation. The same phenomenon applies to each measurement taken in the course of constructing a calibration curve, causing a http://www.chem.utoronto.ca/coursenotes/analsci/stats/ErrRegr.html variation in the slope and intercept of the calculated regression line. This can be reduced - though never completely eliminated - by making replicate measurements for each standard. Multiple calibrations with single values compared to the mean of all three trials. Note how all the regression lines pass close to the centroid of the data. Even with this precaution, we still error in need some way of estimating the likely error (or uncertainty) in the slope and intercept, and the corresponding uncertainty associated with any concentrations determined using the regression line as a calibration function. Tips & links: Skip to uncertainty of the regression Skip to uncertainty of the slope Skip to uncertainty of the intercept Skip to the suggested exercise Skip to Using Excel’s error in slope functions Download a specimen Excel file for this section Navigation: Introduction Bibliography Contact Info Copyright How to Use Concept Map Site Map Excel™ Basics Entering Data Formulas Plotting Functions Trendlines Basic Statistics Stats in Anal Chem Mean and Variance Error and Residuals Probability Confidence Levels Degrees of Freedom Linear Regression Calibration Correlation Linear Portions Regression Equation Regression Errors Using the Calibration Limits of Detection Outliers in Regression Evaluation & Comparison Hypotheses t-test 1- and 2-tailed Tests F-test Summary Quick Links: Site Map Concept Map Next Page Previous Page Next Topic Previous Topic The Uncertainty of the Regression: We saw earlier that the spread of the actual calibration points either side of the line of regression of y on x (which we are using as our calibration function) can be expressed in terms of the regression residuals, (yi − ): The greater these resdiuals, the greater the uncertainty in where the true regression line actually lies. The uncertainty in the regression is therefore calculated in terms of these residuals. Technically, this is the standard error of the regression, sy/x: Note that there are (