Error Of A Summation
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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
Propagation Of Error Division
of a result. We say that "errors in the data propagate through the calculations error propagation formula physics to produce error in the result." 3.2 MAXIMUM ERROR We first consider how data errors propagate through calculations to affect error
Error Propagation Square Root
limits (or maximum error) of results. It's easiest 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 error propagation average and also to indeterminate errors. The underlying mathematics is that 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 error propagation calculator and B respectively. The data quantities are written to show the errors explicitly: [3-1] 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 + &D
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Error Propagation Chemistry
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Adding Errors In Quadrature
Cross Validated Questions Tags Users Badges Unanswered Ask Question _ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm 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 top Standard Error for Sum up vote 1 down vote favorite I am designing an algorithm for a stratified sampling on a population and http://stats.stackexchange.com/questions/164505/standard-error-for-sum then I want to find out what is the error bound for 95% confidence interval, for different sample statistic such as sum of the samples. For this purpose, I need to find Standard Error for sum. I cannot find any information on Standard error other than for mean and proportion. My purpose is SE for sum.. Any information on the formula for SE for SUM, or any links which gives an idea is highly appreciated ! (I have heard SE calculation differs for different sampling techniques, any info about that would be great too). sampling standard-deviation standard-error stratification share|improve this question edited Aug 3 '15 at 17:21 whuber♦ 145k17283541 asked Aug 3 '15 at 15:02 user3218207 61 2 The mean and the sum are usually related by a constant multiple--typically, the mean is $1/n$ times the sum. Since it is obvious that their standard errors will be related by the same multiple, people rarely stop to give explicit formulas for both sums and means: one formula is good enough. –
total obtained by adding a sequence of finite precision floating point numbers, compared to the obvious approach. This is done by keeping a separate running compensation (a variable to accumulate small errors). In particular, simply summing n numbers in sequence has a worst-case https://en.wikipedia.org/wiki/Kahan_summation_algorithm error that grows proportional to n, and a root mean square error that grows as n {\displaystyle {\sqrt {n}}} for random inputs (the roundoff errors form a random walk).[2] With compensated summation, the worst-case error bound is independent of n, so a large number of values can be summed with an error that only depends on the floating-point precision.[2] The algorithm is attributed to William Kahan.[3] Similar, earlier techniques are, for example, Bresenham's line error propagation algorithm, keeping track of the accumulated error in integer operations (although first documented around the same time[4]) and the Delta-sigma modulation[5] (integrating, not just summing the error). Contents 1 The algorithm 1.1 Worked example 2 Accuracy 3 Alternatives 4 Possible invalidation by compiler optimization 5 Support by libraries 6 See also 7 References 8 External links The algorithm[edit] In pseudocode, the algorithm is: function KahanSum(input) var sum = 0.0 var c = 0.0 // error of a A running compensation for lost low-order bits. for i = 1 to input.length do var y = input[i] - c // So far, so good: c is zero. var t = sum + y // Alas, sum is big, y small, so low-order digits of y are lost. c = (t - sum) - y // (t - sum) cancels the high-order part of y; subtracting y recovers negative (low part of y) sum = t // Algebraically, c should always be zero. Beware overly-aggressive optimizing compilers! next i // Next time around, the lost low part will be added to y in a fresh attempt. return sum Worked example[edit] This example will be given in decimal. Computers typically use binary arithmetic, but the principle being illustrated is the same. Suppose we are using six-digit decimal floating point arithmetic, sum has attained the value 10000.0, and the next two values of input(i) are 3.14159 and 2.71828. The exact result is 10005.85987, which rounds to 10005.9. With a plain summation, each incoming value would be aligned with sum and many low order digits lost (by truncation or rounding). The first result, after rounding, would be 10003.1. The second result would be 10005.81828 before rounding, and 10005.8 after rounding. This is not correct. However, with compensated summation, we get the correct rounded result