Error Propagation Standard Deviation Average
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The approach to uncertainty analysis that has been followed up to this point in the discussion has been what is called a top-down approach. Uncertainty components are estimated from direct repetitions of the measurement result. To contrast this with a propagation of error approach, consider the error propagation vs standard deviation simple example where we estimate the area of a rectangle from replicate measurements of length
Error Analysis Standard Deviation
and width. The area $$ area = length \cdot width $$ can be computed from each replicate. The standard deviation of the error propagation mean reported area is estimated directly from the replicates of area. Advantages of top-down approach This approach has the following advantages: proper treatment of covariances between measurements of length and width proper treatment of unsuspected sources of
Error Propagation Covariance
error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period independence from propagation of error model Propagation of error approach combines estimates from individual auxiliary measurements The formal propagation of error approach is to compute: standard deviation from the length measurements standard deviation from the width measurements and combine the two into a standard deviation for area using the approximation for products of two variables how to find propagation of error (ignoring a possible covariance between length and width), $$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$ Exact formula Goodman (1960) derived an exact formula for the variance between two products. Given two random variables, \(x\) and \(y\) (correspond to width and length in the above approximate formula), the exact formula for the variance is: $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$ with \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula) \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to \(s^2\) for width and length, respectively, in the approximate formula) \( E_{ij} = {(\Delta x)^i, (\Delta y)^j}\) where \( \Delta x = x - X \) and \( \Delta y = y - Y \) \( Cov((\Delta x)^2, (\Delta y)^2) = E_{22} - V(x) V(y) \) To obtain the standard deviation, simply take the square root of the above formula. Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation. Approximate formula assumes indpendence The approximate formula assumes that length and
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Propagation Of Error Calculation Example
Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data error propagation mean value 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 are voted up and rise to the http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm 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 K). Now I have two values, that differ slighty and http://stats.stackexchange.com/questions/71419/average-over-two-variables-why-do-standard-error-of-mean-and-error-propagation 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 interpretation of that was always that the manufacturer did a lot of measurements with a calibrated source and calculated the 'descri
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