Counting Error Propagation
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"change" in the value of that quantity. Results are is obtained by mathematical operations on the data, and small changes in any data quantity can affect the value of a result. We error propagation example say that "errors in the data propagate through the calculations to produce error in the result." error propagation division 3.2 MAXIMUM ERROR We first consider how data errors propagate through calculations to affect error limits (or maximum error) of results. It's easiest error propagation physics to first consider determinate errors, which have explicit sign. This leads 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 error propagation calculus 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, 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]
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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 two quantities. In that case the error in the result is the difference in the errors. Summarizing: Sum and difference rule. When two quantities are added (or subtracted), their determinate errors add (or subtract). Now consider multiplication: R = AB. With errors explicitly included: R + ΔR = (A + ΔA)(B + ΔB) = AB + (ΔA)B + A(ΔB) + (ΔA)(ΔB) [3-3] or : ΔR = (ΔA)B + A(ΔB) + (ΔA)(ΔB) This doesn't look like a simple rule. However, when we ex
Detection Introduction Counting Statistics Introduction > > Error Propagation Gas-Filled Detectors Neutron Detection Gamma-Ray Spectroscopy Semiconductor Detectors Detector Lab Error Propagation For all quantities other
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than the directly detected number of counts, we must determine the error propagation chemistry uncertainty by propagating the errors of any operations made. This would include all of the following error propagation log types of operations: adding two counts dividing by a constant subtracting off a background count rate multiplication or division of counts producing a mean value of multiple individual https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm counts combining independent measurements with unequal errors General error propagation can be complicated; however, for the most commonly occurring operations only a few simple relationships are needed. When adding (or subtracting) two measurements each with a given uncertainty, the uncertainty of the final quantity will be produced by summing the errors of the individual parts in http://nsspi.tamu.edu/nssep/courses/basic-radiation-detection/counting-statistics/error-propagation/error-propagation quadrature. Thus, u = x - y σu = √σx2 + σy2 If we divide (or multiply) a quantity with an uncertainty by a constant (i.e., a number with no uncertainty), then the standard deviation of the product is simply equal to the standard deviation of the original value multiplied (or divided) by the constant. Thus, u = Ax σu = Aσx If we multiply two quantities together, then the fractional standard deviation of the product is determined by summing the fractional standard deviations of the original quantities in quadrature. Thus, u = x - y (σu / u ) = √(σx / x)2 + (σy / y)2 Larger Format Watch on YouTube See Script Page 9 / 54 <
it. In doing this it is crucial to understand that all measurements of physical quantities are subject to uncertainties. It is never possible to measure anything exactly. It is good, of course, to http://teacher.nsrl.rochester.edu/phy_labs/AppendixB/AppendixB.html make the error as small as possible but it is always there. And in order to draw valid conclusions the error must be indicated and dealt with properly. Take the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8", how accurate is our result? Well, the height of a person depends on how straight she stands, whether she just got up error propagation (most people are slightly taller when getting up from a long rest in horizontal position), whether she has her shoes on, and how long her hair is and how it is made up. These inaccuracies could all be called errors of definition. A quantity such as height is not exactly defined without specifying many other circumstances. Even if you could precisely specify the "circumstances," your result would still have an counting error propagation error associated with it. The scale you are using is of limited accuracy; when you read the scale, you may have to estimate a fraction between the marks on the scale, etc. If the result of a measurement is to have meaning it cannot consist of the measured value alone. An indication of how accurate the result is must be included also. Indeed, typically more effort is required to determine the error or uncertainty in a measurement than to perform the measurement itself. Thus, the result of any physical measurement has two essential components: (1) A numerical value (in a specified system of units) giving the best estimate possible of the quantity measured, and (2) the degree of uncertainty associated with this estimated value. For example, a measurement of the width of a table would yield a result such as 95.3 +/- 0.1 cm. Significant Figures The significant figures of a (measured or calculated) quantity are the meaningful digits in it. There are conventions which you should learn and follow for how to express numbers so as to properly indicate their significant figures. Any digit that is not zero is significant. Thus 549 has three significant figures and 1.892 has four significant figures. Zeros between non zero digits are significant.