Additive Error Calculations
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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 addition error propagation your answer for the combined result of these measurements and their uncertainties scientifically? additive measure The answer to this fairly common question depends on how the individual measurements are combined in the result. We additive equation will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y, and Z, with uncertainties dX, dY, and dZ, and your final result, R,
Propagation Of Error Division
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. 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 error propagation physics 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 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
"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 error propagation square root the value of a result. We say that "errors in the data propagate
Error Propagation Chemistry
through the calculations to produce error in the result." 3.2 MAXIMUM ERROR We first consider how data errors propagate through calculations
Error Propagation Average
to affect error 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 http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm to other error measures 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 https://www.lhup.edu/~dsimanek/scenario/errorman/propagat.htm to represent the errors in A 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.
in measuring the time required for a weight to fall to the floor, a random error will occur when an experimenter attempts to push a button that starts a timer simultaneously with the http://felix.physics.sunysb.edu/~allen/252/PHY_error_analysis.html release of the weight. If this random error dominates the fall time measurement, then if we repeat the measurement many times (N times) and plot equal intervals (bins) of the fall time ti on the horizontal axis against the number of times a given fall time ti occurs on the vertical axis, our results (see histogram below) should approach an ideal bell-shaped curve (called error propagation a Gaussian distribution) as the number of measurements N becomes very large. The best estimate of the true fall time t is the mean value (or average value) of the distribution: átñ = (SNi=1 ti)/N . If the experimenter squares each deviation from the mean, averages the squares, and takes the square root of that average, the result is a quantity called the "root-mean-square" or additive error calculations the "standard deviation" s of the distribution. It measures the random error or the statistical uncertainty of the individual measurement ti: s = Ö[SNi=1(ti - átñ)2 / (N-1) ].
About two-thirds of all the measurements have a deviation less than one s from the mean and 95% of all measurements are within two s of the mean. In accord with our intuition that the uncertainty of the mean should be smaller than the uncertainty of any single measurement, measurement theory shows that in the case of random errors the standard deviation of the mean smean is given by: sm = s / ÖN , where N again is the number of measurements used to determine the mean. Then the result of the N measurements of the fall time would be quoted as t = átñ ± sm. Whenever you make a measurement that is repeated N times, you are supposed to calculate the mean value and its standard deviation as just described. For a large number of measurements this procedure is somewhat tedious. If you have a calculator with statistical functions it may do the job for you. There is also a simplifie