Propagation Error Calculating Standard Deviation
Contents |
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 division propagation of error approach, consider the simple example where we estimate the area propagation of errors physics of a rectangle from replicate measurements of length and width. The area $$ area = length \cdot width $$ can
Error Propagation Chemistry
be computed from each replicate. The standard deviation of the reported area is estimated directly from the replicates of area. Advantages of top-down approach This approach has the following advantages: proper treatment of
Error Propagation Excel
covariances between measurements of length and width proper treatment of unsuspected sources of 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 error propagation calculus measurements and combine the two into a standard deviation for area using the approximation for products of two variables (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. Als
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 combined result of
Error Propagation Average
these measurements and their uncertainties scientifically? The answer to this fairly common question depends error propagation definition on how the individual measurements are combined in the result. We will treat each case separately: Addition of measured quantities If propagation of errors pdf 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, then the uncertainty dR http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm 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 = 14.4 m - 9.3 m = http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm 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 error in X with the absolute value of the constant, and this will g
Community Forums > Mathematics > Set Theory, Logic, Probability, Statistics > We've just passed 300 Insights! View them here! What a resource! Dismiss Notice Dismiss Notice Join Physics Forums Today! The friendliest, high quality science and math community on the planet! Everyone who loves science is here! https://www.physicsforums.com/threads/error-propagation-with-averages-and-standard-deviation.608932/ Error propagation with averages and standard deviation Page 1 of 2 1 2 Next > May http://www.chem.hope.edu/~polik/Chem345-2000/errorpropagation.htm 25, 2012 #1 rano I was wondering if someone could please help me understand a simple problem of error propagation going from multiple measurements with errors to an 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 error propagation 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 associated with the mass of each rock. propagation of error 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 an 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 lo
x, y, or z leads to an error in the determination of u. This is simply the multi-dimensional definition of slope. It describes how changes in u depend on changes in x, y, and z. Example: A miscalibrated ruler results in a systematic error in length measurements. The values of r and h must be changed by +0.1 cm. 3. Random Errors Random errors in the measurement of x, y, or z also lead to error in the determination of u. However, since random errors can be both positive and negative, one should examine (du)2 rather than du. If the measured variables are independent (non-correlated), then the cross-terms average to zero as dx, dy, and dz each take on both positive and negative values. Thus, Equating standard deviation with differential, i.e., results in the famous error propagation formula This expression will be used in the Uncertainty Analysis section of every Physical Chemistry laboratory report! Example: There is 0.1 cm uncertainty in the ruler used to measure r and h. Thus, the expected uncertainty in V is ±39 cm3. 4. Purpose of Error Propagation · Quantifies precision of results Example: V = 1131 ± 39 cm3 · Identifies principle source of error and suggests improvement Example: Determine r better (not h!) · Justifies observed standard deviation If sobserved » scalculated then the observed standard deviation is accounted for If sobserved differs significantly from scalculated then perhaps unrealistic values were chosen for sx, sy, and sz. · Identifies type of error If ½uobserrved - uliterature½ £ scalculated then error is random error If ½uobserrved - uliterature½ >> scalculated then error is systematic error 5. Calculating and Reporting Values when using Error Propagation Use full precision (keep extra significant figures and do not round) until the end of a calculation. Then keep two significant figures for the uncertainty and match precision for the value. Example: V = 1131 ± 39 cm3 6. Comparison of Error Propagation to Significant Figures Use of significant figures in calculations is a rough estimate of error propagation. Example: Keeping two significant figures in this example implies a result of V = 1100 ± 100 cm3, which is much less precise than the result of V = 1131 ± 39 cm3 derived by error propagat