Propagated Error Formula
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Error Propagation Physics
Nature Expand/collapse global location Propagation of Error Last updated 20:33, 14 May 2016 Save as PDF Share Share Share Tweet Share
Error Propagation Chemistry
IntroductionDerivation of Exact FormulaDerivation of Arithmetic ExampleCaveats and WarningsDisadvantages of Propagation of Error ApproachTreatment of Covariance TermsReferencesContributors Propagation of Error (or Propagation of Uncertainty) is defined as the effects on a function by a
Error Propagation Definition
variable's uncertainty. It is a calculus derived statistical calculation designed to combine uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. Introduction Every measurement has an air of uncertainty about it, and not all uncertainties are equal. Therefore, the ability to properly combine uncertainties from different measurements is crucial. Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, error propagation excel time of day, etc. Typically, error is given by the standard deviation (\(\sigma_x\)) of a measurement. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. For example, lets say we are using a UV-Vis Spectrophotometer to determine the molar absorptivity of a molecule via Beer's Law: A = ε l c. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must be applied to measure a more exact uncertainty of the molar absorptivity. This example will be continued below, after the derivation (see Example Calculation). Derivation of Exact Formula Suppose a certain experiment requires multiple instruments to carry out. These instruments each have different variability in their measurements. The results of each instrument are given as: a, b, c, d... (For simplification purposes, only the variables a, b, and c will be used throughout this derivation). The end result desired is \(x\), so that \(x\) is dependent on a, b, and c. It can be written that \(x\) is a function of these variables: \[x=f(a,b,c) \tag{1}\] Because each measurement has an uncertainty about its mean, it can be written that the uncertainty of dxi
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 these measurements and their uncertainties scientifically? The answer to this error propagation square root fairly common question depends on how the individual measurements are combined in the result. We error propagation inverse will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y, and Z, with uncertainties dX, error propagation average dY, and dZ, and your final result, R, is the sum or difference of these quantities, then the uncertainty dR is: Here the upper equation is an approximation that can also serve as an upper bound for the error. http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_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 = 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 http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm 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 give you the error in R: If you compare this to the above rule for multiplication of two quantities, you see that this is just the special case of that rule for the uncertainty in c, dc = 0. Example: If an object is realeased from rest and is in free fall, and if you measure the velocity of t
WorkSocial MediaSoftwareProgrammingWeb Design & DevelopmentBusinessCareersComputers Online Courses B2B Solutions Shop for Books San Francisco, CA Brr, it´s cold outside Search Submit RELATED ARTICLES Simple Error Propagation Formulas for Simple Expressions Key Concepts in Human Biology and Physiology Chronic Pain and Individual Differences in Pain Perception Pain-Free and Hating It: Peripheral Neuropathy Neurotransmitters That http://www.dummies.com/education/science/biology/simple-error-propagation-formulas-for-simple-expressions/ Reduce or Block Pain Load more EducationScienceBiologySimple Error Propagation Formulas for Simple Expressions Simple Error Propagation Formulas for Simple Expressions Related Book Biostatistics For Dummies By John Pezzullo Even though some general error-propagation formulas are very complicated, the rules for propagating SEs through some simple mathematical expressions are much easier to work with. Here are some of the most common simple rules. All the rules that involve two or more variables assume that those variables error propagation have been measured independently; they shouldn't be applied when the two variables have been calculated from the same raw data. Adding or subtracting a constant doesn't change the SE Adding (or subtracting) an exactly known numerical constant (that has no SE at all) doesn't affect the SE of a number. So if x = 38 ± 2, then x + 100 = 138 ± 2. Likewise, if x = 38 ± 2, then x - 15 propagated error formula = 23 ± 2. Multiplying (or dividing) by a constant multiplies (or divides) the SE by the same amount Multiplying a number by an exactly known constant multiplies the SE by that same constant. This situation arises when converting units of measure. For example, to convert a length from meters to centimeters, you multiply by exactly 100, so a length of an exercise track that's measured as 150 ± 1 meters can also be expressed as 15,000 ± 100 centimeters. For sums and differences: Add the squares of SEs together When adding or subtracting two independently measured numbers, you square each SE, then add the squares, and then take the square root of the sum, like this: For example, if each of two measurements has an SE of ± 1, and those numbers are added together (or subtracted), the resulting sum (or difference) has an SE of A useful rule to remember is that the SE of the sum or difference of two equally precise numbers is about 40 percent larger than the SE of one of the numbers. When two numbers of different precision are combined (added or subtracted), the precision of the result is determined mainly by the less precise number (the one with the larger SE). If one number has an SE of ± 1 and another has an SE of ± 5
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