Error Propagation Exp
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constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -. RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS) SUM RULE: error propagation example When R = A + B then ΔR = ΔA + ΔB error propagation division DIFFERENCE RULE: When R = A - B then ΔR = ΔA - ΔB PRODUCT RULE: When R error propagation physics = AB then (ΔR)/R = (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 = n(ΔA)/A
Error Propagation Calculus
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 used, when appropriate. We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it error propagation khan academy 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 = 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
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 your answer for the combined result of these measurements and their uncertainties scientifically? The error propagation average answer to this fairly common question depends on how the individual measurements are combined in the
Error Propagation Chemistry
result. We will treat each case separately: Addition of measured quantities If you have measured values for the quantities X, Y, and Z,
Error Propagation Log
with uncertainties dX, dY, and dZ, 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 https://www.lhup.edu/~dsimanek/scenario/errorman/rules.htm 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 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 http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm 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, and you need to multiply it with a constant that is known exactly? What is the error then? This is easy: just multiply the error in X with the absolute value of the constant, and this will give you the error in R: If you compare this to the above rule for multiplication of two quantities, you see that this is just the special case of that rule for the uncertainty in c, dc = 0. Example: If an object is realeased fr
EXP, LNe, LOG, SIN, error propagation SQR, TAN. Variables are one or two characters, e.g. I, X, df, X7. Nested parentheses are useful, e.g. ((X+Y)*Z). No implicit multiplication, error propagation exp e.g. ((X+Y)Z) is not allowed. Variables are not case sensitive: x=X. Scientific notation: 1.23x10-3 is written as 1.23E-3. Enter your equation without an "=" sign. -----Example: ----- To evaluate K2, knowing K1, H, R, T2,and T1 in the equation: ln(K2/K1) = - H/R( 1/T2 - 1/T1) Solve for K2. You would then enter Equation: K1*EXP(-H/R*(1/T2-1/T1)) Equation: Result= Colby College Chemistry, T. W. Shattuck