Error Propagation Chemistry
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Engineering Medicine Agriculture Photosciences Humanities Periodic Table of the Elements Reference Tables Physical Constants Units and Conversions Organic Chemistry Glossary Search site Search Search Go back to previous article Username Password Sign in Sign in error propagation physics Sign in Registration Forgot password Expand/collapse global hierarchy Home Core Analytical Chemistry Quantifying Nature error analysis chemistry Expand/collapse global location Propagation of Error Last updated 20:33, 14 May 2016 Save as PDF Share Share Share Tweet Share IntroductionDerivation standard deviation chemistry 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 variable's uncertainty. error propagation rules chemistry 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, time of day,
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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 of the ith measurement
Engineering Medicine Agriculture Photosciences Humanities Periodic Table of the Elements Reference Tables Physical Constants Units and Conversions Organic Chemistry Glossary Search site Search Search Go back to previous
Error Propagation Example
article Username Password Sign in Sign in Sign in Registration Forgot password error propagation division Expand/collapse global hierarchy Home Textbook Maps Analytical Chemistry Textbook Maps Map: Analytical Chemistry 2.0 (Harvey) 4: Evaluating Analytical Data error propagation calculus Expand/collapse global location 4.3: Propagation of Uncertainty Last updated 10:52, 25 May 2016 Save as PDF Share Share Share Tweet Share 4.3.1 A Few Symbols4.3.2 Uncertainty When Adding or Subtracting4.3.3 Uncertainty http://chem.libretexts.org/Core/Analytical_Chemistry/Quantifying_Nature/Significant_Digits/Propagation_of_Error When Multiplying or Dividing4.3.4 Uncertainty for Mixed Operations4.3.5 Uncertainty for Other Mathematical Functions4.3.6 Is Calculating Uncertainty Actually Useful?Contributors Suppose you dispense 20 mL of a reagent using the Class A 10-mL pipet whose calibration information is given in Table 4.9. If the volume and uncertainty for one use of the pipet is 9.992 ± 0.006 mL, what is the volume and uncertainty when http://chem.libretexts.org/Textbook_Maps/Analytical_Chemistry_Textbook_Maps/Map%3A_Analytical_Chemistry_2.0_(Harvey)/04_Evaluating_Analytical_Data/4.3%3A_Propagation_of_Uncertainty we use the pipet twice? As a first guess, we might simply add together the volume and the maximum uncertainty for each delivery; thus \[\mathrm{(9.992\: mL + 9.992\: mL) ± (0.006\: mL + 0.006\: mL) = 19.984 ± 0.012\: mL}\] It is easy to appreciate that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that with each use the indeterminate error is in the same direction and is as large as possible. At the other extreme, we might assume that the uncertainty for one delivery is positive and the other is negative. If we subtract the maximum uncertainties for each delivery, \[\mathrm{(9.992\: mL + 9.992\: mL) ± (0.006\: mL - 0.006\: mL) = 19.984 ± 0.000\: mL}\] we clearly underestimate the total uncertainty. So what is the total uncertainty? From the previous discussion we know that the total uncertainty is greater than ±0.000 mL and less than ±0.012 mL. To estimate the cumulative effect of multiple uncertainties we use a mathematical technique known as the propagation of uncertainty. Our treatment of the
is important to work as accurately and precisely as possible. Therefore, almost all analytical, volumetric glassware shows the error that is made when using the glassware, such that you can calculate the size of the error in the experiment. An example is given in http://webchem.science.ru.nl/chemical-analysis/error-propagation/ the picture below, which shows a close-up of a 100 mL volumetric flask. The error that you make when using this flask is ±0.1 mL. In the remainder of this section, we will learn what this actually means and how it influences a final experimental result. (Source: Wikipedia) Question: is this a random or systematic error? More on volumetric glassware The error displayed on volumetric glassware is the random error resulting from the error propagation production process. In the case of the volumetric flask above, this would mean that a collection of identical flasks together has an error of ±0.1 mL (in other words: the standard deviation is 0.1 mL). However, individual flasks from the collection may have an error of +0.05 mL or -0.07 mL (Question: are these systematic or random errors?). For accurate results, you should constantly use different glassware such that errors cancel out. A second error propagation chemistry option is to calibrate the glassware: determine the volume by weighing. The error after calibration should be much smaller than the error shown on the glassware. Moreover, this error has now become random instead of systematic! Since this requires a lot of work each time you want to use volumetric glassware, we will from now on assume that errors shown on volumetric glassware are random errors. For example, each time when using the depicted volumetric flask properly, the volume will be 100 mL with an error of ±0.1 mL. Significant figures As a general rule, the last reported figure of a result is the first with uncertainty. Assume that we have measured the weight of an object: 80 kg. To indicate that we are not sure of the last digit,we can write 80 ± 1 kg. If we would have used a better scale to weigh the object, we might have found 80.00 ± 0.01 kg. Question: is the second result more precise or more accurate than the first? We can also display the error in a relative way. For instance, 80 ± 1 kg is identical to 80 ± 1.25%. The order of magnitude of the result should be as clear as possible. Therefore, the preferred notation of for instance 0.0174 ± 0.0002 is (1.74 ± 0.02