Error Calculation Multiplication
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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 your error propagation division answer for the combined result of these measurements and their uncertainties scientifically? The
Error Propagation Division Example
answer to this fairly common question depends on how the individual measurements are combined in the result. We will
Propagation Of Uncertainty Multiplication And Division
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, is
Error Calculation Formula
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 as x2 = mental calculation multiplication 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, and you need to mult
find that the error in this measurement is 0.001 in. To find the area we multiply the width (W) and length (L). The area then is L x W how to calculate error when multiplying = (1.001 in) x (1.001 in) = 1.002001 in2 which rounds to 1.002 propagation of error physics in2. This gives an error of 0.002 if we were given that the square was exactly super-accurate 1 inch a side. error propagation calculator This is an example of correlated error (or non-independent error) since the error in L and W are the same. The error in L is correlated with that of in W. Now, suppose http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm that we made independent determination of the width and length separately with an error of 0.001 in each. In this case where two independent measurements are performed the errors are independent or uncorrelated. Therefore the error in the result (area) is calculated differently as follows (rule 1 below). First, find the relative error (error/quantity) in each of the quantities that enter to the calculation, relative error in width is http://www.utm.edu/~cerkal/Lect4.html 0.001/1.001 = 0.00099900. The resultant relative error is Relative Error in area = Therefore the absolute error is (relative error) x (quantity) = 0.0014128 x 1.002001=0.001415627. which rounds to 0.001. Therefore the area is 1.002 in2± 0.001in.2. This shows that random relative errors do not simply add arithmetically, rather, they combine by root-mean-square sum rule (Pythagorean theorem). Let’s summarize some of the rules that applies to combining error when adding (or subtracting), multiplying (or dividing) various quantities. This topic is also known as error propagation. 2. Error propagation for special cases: Let σx denote error in a quantity x. Further assume that two quantities x and y and their errors σx and σy are measured independently. In this case relative and percent errors are defined as Relative error = σx / x, Percent error = 100 (σx / x) Multiplying or dividing with a constant. The resultant absolute error also is multiplied or divided. Multiplication or division, relative error. Addition or subtraction: In this case, the absolute errors obey Pythagorean theorem. If a and b are constants, If there are more than two measured quantities, you can extend expressions provided above by
dividing Is one result consistent with another? What if there are several measurements of the same quantity? How can one estimate the uncertainty of a slope on a graph? Uncertainty in a http://spiff.rit.edu/classes/phys273/uncert/uncert.html single measurement Bob weighs himself on his bathroom scale. The smallest divisions on the scale are 1-pound marks, so the least count of the instrument is 1 pound. Bob reads his weight as closest to the 142-pound mark. He knows his weight must be larger than 141.5 pounds (or else it would be closer to the 141-pound mark), but error propagation smaller than 142.5 pounds (or else it would be closer to the 143-pound mark). So Bob's weight must be weight = 142 +/- 0.5 pounds In general, the uncertainty in a single measurement from a single instrument is half the least count of the instrument. Fractional and percentage uncertainty What is the fractional uncertainty in Bob's weight? uncertainty in weight error propagation division fractional uncertainty = ------------------------ value for weight 0.5 pounds = ------------- = 0.0035 142 pounds What is the uncertainty in Bob's weight, expressed as a percentage of his weight? uncertainty in weight percentage uncertainty = ----------------------- * 100% value for weight 0.5 pounds = ------------ * 100% = 0.35% 142 pounds Combining uncertainties in several quantities: adding or subtracting When one adds or subtracts several measurements together, one simply adds together the uncertainties to find the uncertainty in the sum. Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is 147 +/- 3 cm tall. If Jane stands on top of Dick's head, how far is her head above the ground? combined height = 186 cm + 147 cm = 333 cm uncertainty in combined height = 2 cm + 3 cm = 5 cm combined height = 333 cm +/- 5 cm Now, if all the quantities have roughly the same magnitude and uncertainty -- as in the example above -- the result makes perfect sense. But if one tr
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