Propagation Error Rules
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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
Error Propagation Inverse
you state your answer for the combined result of these measurements and error propagation calculator their uncertainties scientifically? The answer to this fairly common question depends on how the individual measurements are combined in
Error Propagation Square Root
the result. We 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, error propagation physics 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. Please note that the rule is the same for addition and subtraction of quantities. Example: Suppose we have measured the starting position as x1 error propagation chemistry = 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 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
constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -. RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS) SUM RULE: When R = A + B then ΔR = ΔA + ΔB DIFFERENCE RULE: When R = A -
Error Propagation Average
B then ΔR = ΔA - ΔB PRODUCT RULE: When R = AB then (ΔR)/R = error propagation excel (ΔA)/A + (ΔB)/B QUOTIENT RULE: When R = A/B then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R =
Error Propagation Definition
n(ΔA)/A or (ΔR) = n An-1(ΔA) Memory clues: When quantities are added (or subtracted) their absolute errors add (or subtract). But when quantities are multiplied (or divided), their relative fractional errors add (or subtract). These rules will be freely http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm used, when appropriate. We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it must be expressed in radians. RULES FOR ELEMENTARY FUNCTIONS (DETERMINATE ERRORS) EQUATION ERROR EQUATION R = sin q ΔR = (dq) cos q R = cos q ΔR = -(dq) sin q R = tan q ΔR = (dq) sec2 q R = ex ΔR = (Δx) ex R = e-x ΔR = -(Δx) e-x R = https://www.lhup.edu/~dsimanek/scenario/errorman/rules.htm ln(x) ΔR = (Δx)/x Any measures of error may be converted to relative (fractional) form by using the definition of relative error. The fractional error in x is: fx = (ΔR)x)/x where (ΔR)x is the absolute ereror in x. Therefore xfx = (ΔR)x. The rules for indeterminate errors are simpler. RULES FOR ELEMENTARY OPERATIONS (INDETERMINATE ERRORS) SUM OR DIFFERENCE: When R = A + B then ΔR = ΔA + ΔB PRODUCT OR QUOTIENT: When R = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) The indeterminate error rules for elementary functions are the same as those for determinate errors except that the error terms on the right are all positive. Students who are taking calculus will notice that these rules are entirely unnecessary. The determinate error equations may be found by differentiating R, then replading dR, dx, dy, etc. with ΔR, Δx, Δy, etc. This is equivalent to expanding ΔR as a Taylor series, then neglecting all terms of higher order than 1. This is a valid approximation when (ΔR)/R, (Δx)/x, etc. are all small fractions. The indeterminate error equations may be constructed from the determinate error equations by algebraically reaarranging the final resultl into standard form: ΔR = ( )Δx + ( )Δy + ( )Δz . The coefficients in parantheses ( ), and/or the errors th
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