Error Propagation When Taking Average
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"change" in the value of that quantity. Results are is obtained by mathematical operations on the data, and small error propagation average standard deviation changes in any data quantity can affect the value of a result.
Error Propagation Mean
We say that "errors in the data propagate through the calculations to produce error in the result." 3.2 MAXIMUM how to find error propagation 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 error propagation mean value 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 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,
Calculating Error Propagation
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 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 t
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Propagation Of Error Division
site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm The best answers are voted up and rise to the top Error propagation on weighted mean up vote 1 down vote favorite I understand that, if errors are random and independent, the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the error of $x + y$ is given by $\sqrt{e_1^2 + e_2^2}$, http://math.stackexchange.com/questions/123276/error-propagation-on-weighted-mean where $e_1$ and $e_2$ and the errors of $x$ and $y$, respectively. However, I have not yet been able to find how to calculate the error of both the arithmetic mean and the weighted mean of the two measured quantities. How do errors propagate in these cases? statistics error-propagation share|cite|improve this question edited Mar 22 '12 at 17:02 Michael Hardy 158k15145350 asked Mar 22 '12 at 13:46 plok 10815 add a comment| 2 Answers 2 active oldest votes up vote 3 down vote accepted The first assertion assumes one takes mean squared errors, which in probabilistic terms translates into standard deviations. Now, probability says that the variance of two independent variables is the sum of the variances. Hence, if $z = x + y$ , $\sigma_z^2 = \sigma_x^2 + \sigma_y^2 $ and $$e_z = \sigma_z = \sqrt{\sigma_x^2 + \sigma_y^2} = \sqrt{e_x^2 + e_y^2} $$ Knowing this, and knowing that $Var(a X) = a^2 Var(X)$, if $z = a x + (1-a) y$ (weighted mean, if $ 0\le a \le1$) we get: $$\sigma_z^2 = a^2\sigma_x^2 + (1-a)^2\sigma_y^2 $$ $$e_z = \sqrt{a^2 e_x^2 + (1-a)^2 e_y^2} = a \sqrt{ e_x^2 + \left(\frac{1-a}{a}\right)^2 e_y^2} $$ In particular, if $a=1/2$ , then $e_z = \frac{1}{2}\sq
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 http://stats.stackexchange.com/questions/48948/propagation-of-uncertainty-through-an-average Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with 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 error propagation question Anybody can answer The best answers are voted up and rise to the top Propagation of uncertainty through an average up vote 1 down vote favorite I have a set of distance measurements that are all accurate to +/- 0.01 M. {1.00,2.00,3,00} We can obtain the distance moved between measurements by saying {2-1, 3-2} its trivial to see we moved 1M each time. My question is this. error propagation mean If you want to know the average distance moved, how you you carry the +/- 0.01M through the average. I would like to report the Average Difference +/- the uncertainty. How do I calculate the uncertainty? (My real data is more messy than this). standard-error error uncertainty error-propagation share|improve this question edited Jan 31 '13 at 7:55 mpiktas 24.7k449104 asked Jan 31 '13 at 6:28 MARCO HOWARD 61 add a comment| 1 Answer 1 active oldest votes up vote 1 down vote Ok, there are two issues here. The first is the general question of how to use known uncertainty in estimating the mean and variance. The second is the specific issue relating to the fact that you are taking differences. In general: In a more general situation, one might have to average a number of measurements each with known standard error $\sigma$. In which case the total variance is the sum of the sample variance and the measurement variance. This is analogous to ANOVA where there is the total variance is the sum of the between groups and within groups variance. Imagine each measurement was actually a little subsample group of repeated measurements, then this is exactly what you would