Error Propagation When Taking An Average
Contents |
"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 affect the value of a result. We error propagation average standard deviation say that "errors in the data propagate through the calculations to produce error in the
Error Propagation Weighted Average
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 error propagation mean easiest to first consider determinate errors, which have explicit sign. This leads to useful rules for error propagation. Then we'll modify and extend the rules to other error measures and also to indeterminate errors. The underlying mathematics
How To Find Error Propagation
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 ΔA and ΔB to represent the errors in A and B respectively. The data quantities are written to show the errors error propagation mean value 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). Now consider multiplication: R = AB. With errors explicitly included: R + ΔR = (A + ΔA)(B + ΔB) = AB + (ΔA)B + A(ΔB) + (ΔA)(ΔB) [3-3] or : ΔR = (ΔA)B + A(ΔB) + (ΔA)(ΔB) This doesn't look like a simple rule
or more quantities, each with their individual uncertainties, and then combine the information from these quantities in order to come up with a final result of our experiment. How can you state your answer for the
Error Propagation Example
combined result of these measurements and their uncertainties scientifically? The answer to this fairly
Error Propagation Division
common question depends on how the individual measurements are combined in the result. We will treat each case separately: Addition error propagation physics of measured quantities If you have measured values for the quantities X, Y, and Z, with uncertainties dX, dY, and dZ, and your final result, R, is the sum or difference of these quantities, https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm then the uncertainty dR is: Here the upper equation is an approximation that can also serve as an upper bound for the error. Please note that the rule is the same for addition and subtraction of quantities. Example: Suppose we have measured the starting position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. Then the displacement is: Dx = x2-x1 http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm = 14.4 m - 9.3 m = 5.1 m and the error in the displacement is: (0.22 + 0.32)1/2 m = 0.36 m Multiplication of measured quantities In the same way as for sums and differences, we can also state the result for the case of multiplication and division: Again the upper line is an approximation and the lower line is the exact result for independent random uncertainties in the individual variables. And again please note that for the purpose of error calculation there is no difference between multiplication and division. Example: We have measured a displacement of x = 5.1+-0.4 m during a time of t = 0.4+-0.1 s. What is the average velocity and the error in the average velocity? v = x / t = 5.1 m / 0.4 s = 12.75 m/s and the uncertainty in the velocity is: dv = |v| [ (dx/x)2 + (dt/t)2 ]1/2 = 12.75 m/s [(0.4/5.1)2 + (0.1/0.4)2]1/2 = 3.34 m/s Multiplication with a constant What if you have measured the uncertainty in an observable X, and you need to multiply it with a constant that is known exactly? What is the error then? This is easy: just multiply the
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about http://stats.stackexchange.com/questions/71419/average-over-two-variables-why-do-standard-error-of-mean-and-error-propagation Stack Overflow the company Business Learn more about hiring developers or posting ads with http://physics.appstate.edu/undergraduate-programs/laboratory/resources/error-propagation us Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers error propagation are voted up and rise to the top Average over two variables: Why do standard error of mean and error propagation differ and what does that mean? up vote 3 down vote favorite I'm doing an experiment with a cryostat to determine the critical temperature for lead. To avoid asymmetries, I determine the critical temperature both through heating (going from 2 K to 10 K) and cooling (10 K -> 2 error propagation mean K). Now I have two values, that differ slighty and I average them. So a measurement of (6.942 $\pm$ 0.020) K and (6.959 $\pm$ 0.019) K gives me an average of 6.951 K. Now the question is: what is the error of that average? One way to do it would be to calculate the variance of this sample (containing two points), take the square root and divide by $\sqrt{2}$. This gives me an SEM of 0.0085 K, which is too low for my opinion (where does this precision come from?) The other way is to say the the mean is a function of two variables, $\bar{T} = \frac{T_1 + T_2}{2}$, therefore by error propagation the error is $\Delta T = \frac12\sqrt{(\Delta T_1)^2+(\Delta T_2)^2}$, and that gives me a much more rational value of 0.014. I see how those values differ in terms of numbers, but which one is correct when talking about the correct estimate for the standard deviation? mean standard-error measurement-error error-propagation share|improve this question edited Sep 29 '13 at 21:32 gung 74.1k19160309 asked Sep 29 '13 at 21:05 Wojciech Morawiec 1164 @COOLSerdash That's actually another point I have thought about: The numbers after the $\pm$ denote the error of the thermometer, as given by the manufacturer. My interpretat
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 final answer will, of course, have some level of uncertainty. For instance, in lab you 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 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 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. 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