Error Propagation Formula Average
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Community Forums > Mathematics > Set Theory, Logic, Probability, Statistics > Dismiss Notice Join Physics Forums Today! The friendliest, high quality science and math community on the planet! Everyone who loves science is here! Error propagation with averages and standard error propagation formula physics deviation Page 1 of 2 1 2 Next > May 25, 2012 #1 rano I error propagation formula excel was wondering if someone could please help me understand a simple problem of error propagation going from multiple measurements with errors to an
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average incorporating these errors. I have looked on several error propagation webpages (e.g. UC physics or UMaryland physics) but have yet to find exactly what I am looking for. I would like to illustrate my question with some
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example data. Suppose we want to know the mean ± standard deviation (mean ± SD) of the mass of 3 rocks. We weigh these rocks on a balance and get: Rock 1: 50 g Rock 2: 10 g Rock 3: 5 g So we would say that the mean ± SD of these rocks is: 21.6 ± 24.6 g. But now let's say we weigh each rock 3 times each and now there is some error error propagation formula for division associated with the mass of each rock. Let's say that the mean ± SD of each rock mass is now: Rock 1: 50 ± 2 g Rock 2: 10 ± 1 g Rock 3: 5 ± 1 g How would we describe the mean ± SD of the three rocks now that there is some uncertainty in their masses? Would it still be 21.6 ± 24.6 g? Some error propagation websites suggest that it would be the square root of the sum of the absolute errors squared, divided by N (N=3 here). But in this case the mean ± SD would only be 21.6 ± 2.45 g, which is clearly too low. I think this should be a simple problem to analyze, but I have yet to find a clear description of the appropriate equations to use. If my question is not clear please let me know. Any insight would be very appreciated. rano, May 25, 2012 Phys.org - latest science and technology news stories on Phys.org •Game over? Computer beats human champ in ancient Chinese game •Simplifying solar cells with a new mix of materials •Imaged 'jets' reveal cerium's post-shock inner strength May 25, 2012 #2 viraltux rano said: ↑ I was wondering if someone could please help me understand a simple problem of error propagation going from multiple measurements with errors to a
uncertainty of an answer obtained from a calculation. Every time data are measured, there is an uncertainty associated with that measurement. (Refer to guide to Measurement and Uncertainty.) If these measurements used in your calculation have some uncertainty associated with them, then the
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final answer will, of course, have some level of uncertainty. For instance, in lab you general error propagation formula might measure an object's position at different times in order to find the object's average velocity. Since both distance and time measurements have error propagation rules uncertainties associated with them, those uncertainties follow the numbers throughout the calculations and eventually affect your final answer for the velocity of that object. How would you determine the uncertainty in your calculated values? In lab, graphs are https://www.physicsforums.com/threads/error-propagation-with-averages-and-standard-deviation.608932/ often used where LoggerPro software calculates uncertainties in slope and intercept values for you. In other classes, like chemistry, there are particular ways to calculate uncertainties. In fact, since uncertainty calculations are based on statistics, there are as many different ways to determine uncertainties as there are statistical methods. The error propagation methods presented in this guide are a set of general rules that will be consistently used for all levels of physics classes in this department. http://physics.appstate.edu/undergraduate-programs/laboratory/resources/error-propagation In the following examples: q is the result of a mathematical operation δ is the uncertainty associated with a measurement. For example, if you have a measurement that looks like this: m = 20.4 kg ±0.2 kg Thenq = 20.4 kg and δm = 0.2 kg First Step: Make sure that your units are consistent Make sure that you are using SI units and that they are consistent. If you are converting between unit systems, then you are probably multiplying your value by a constant. Please see the following rule on how to use constants. Logger Pro If you are using a curve fit generated by Logger Pro, please use the uncertainty associated with the parameters that Logger Pro give you. In the above linear fit, m = 0.9000 andδm = 0.05774. The uncertainty should be rounded to 0.06, which means that the slope must be rounded to the hundredths place as well: m = 0.90± 0.06 If the above values have units, don't forget to include them. Constants If an expression contains a constant, B, such that q =Bx, then: You can see the the constant B only enters the equation in that it is used to determine q. Example: F = mg = (20.4 kg)(-9.80 m/s2) = -199.92 kgm/s2 δF/F = δm/m δF/(-199.92 kgm/s2) = (0.2 kg)/(20.4 kg) δF = ±1.96 kgm/s2 δF = ±2 kgm/s
"change" in the value of that quantity. Results are is obtained by mathematical operations on the data, and small changes in any data quantity can https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm affect the value of a result. We say that "errors in the data propagate through the calculations to produce error in the result." 3.2 MAXIMUM ERROR We first consider how data errors propagate through calculations to affect error limits (or maximum error) of results. It's easiest to first consider determinate errors, which have explicit sign. This leads to useful rules for error propagation. Then we'll modify and extend error propagation the rules to other error measures and also to indeterminate errors. The underlying mathematics is that of "finite differences," an algebra for dealing with numbers which have relatively small variations imposed upon them. The finite differences we are interested in are variations from "true values" caused by experimental errors. Consider a result, R, calculated from the sum of two data quantities A and B. For this discussion we'll use error propagation formula ΔA and ΔB to represent the errors in A and B respectively. The data quantities are written to show the errors explicitly: [3-1] A + ΔA and B + ΔB We allow the possibility that ΔA and ΔB may be either positive or negative, the signs being "in" the symbols "ΔA" and "ΔB." The result of adding A and B is expressed by the equation: R = A + B. When errors are explicitly included, it is written: (A + ΔA) + (B + ΔB) = (A + B) + (Δa + δb) So the result, with its error ΔR explicitly shown in the form R + ΔR, is: R + ΔR = (A + B) + (Δa + Δb) [3-2] The error in R is: ΔR = ΔA + ΔB. We conclude that the error in the sum of two quantities is the sum of the errors in those quantities. You can easily work out the case where the result is calculated from the difference of two quantities. In that case the error in the result is the difference in the errors. Summarizing: Sum and difference rule. When two quantities are added (or subtracted), their determinate errors add (or subtract).
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