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# Error propagation

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### Error propagation

1. 1. Error Propagation The analysis of uncertainties (errors) in measurements and calculations is essential in the physics laboratory. For example, suppose you measure the length of a long rod by making three measurement x = xbest ± ∆x, y = ybest ± ∆y, and z = zbest ± ∆z. Each of these measurements has its own uncertainty ∆x, ∆y, and ∆z respectively. What is the uncertainty in the length of the rod L = x + y + z? When we add the measurements do the uncertainties ∆x, ∆y, ∆z cancel, add, or remain the same? Likewise , suppose we measure the dimensions b = bbest ± ∆b, h = hbest ± ∆h, and w = wbest ± ∆w of a block. Again, each of these measurements has its own uncertainty ∆b, ∆h, and ∆w respectively. What is the uncertainty in the volume of the block V = bhw? Do the uncertainties add, cancel, or remain the same when we calculate the volume? In order for us to determine what happens to the uncertainty (error) in the length of the rod or volume of the block we must analyze how the error (uncertainty) propagates when we do the calculation. In error analysis we refer to this as error propagation. There is an error propagation formula that is used for calculating uncertainties when adding or subtracting measurements with uncertainties and a different error propagation formula for calculating uncertainties when multiplying or dividing measurements with uncertainties. Let’s first look at the formula for adding or subtracting measurements with uncertainties. Adding or Subtracting Measurements with Uncertainties. Suppose you make two measurements, x = xbest ± ∆x y = ybest ± ∆y What is the uncertainty in the quantity q = x + y or q = x – y? To obtain the uncertainty we will find the lowest and highest probable value of q = x + y. Note that we would like to state q in the standard form of q = qbest ± ∆q where qbest = xbest + ybest. (highest probable value of q = x + y): (xbest+ ∆x) + (ybest + ∆y) = (xbest+ ybest) + (∆x +∆y) = qbest + ∆q (lowest probable value of q = x + y): (xbest- ∆x) + (ybest - ∆y) = (xbest+ ybest) - (∆x +∆y) = qbest – ∆q Thus, we that ∆q = ∆x + ∆y is the uncertainty in q = x + y. A similar result applies if we needed to obtain the uncertainty in the difference q = x – y. If we had added or subtracted more than two 1
2. 2. measurements x, y, ......, z each with its own uncertainty ∆x, ∆y, ......... , ∆z respectively , the result would be ∆q = ∆x + ∆y + ......... + ∆z Now, if the uncertainties ∆x, ∆y, ........., ∆z are random and independent, the result is ∆q = (∆x) 2 + (∆y ) 2 + ......... + (∆z ) 2 Ex. x = 3.52 cm ± 0.05 cm y = 2.35 cm ± 0.04 cm Calculate q = x + y We would like to state q in the standard form of q = qbest ± ∆q xbest = 3.52cm, ∆x = 0.05cm ybest = 2.35cm, ∆y = 0.04cm qbest = xbest + ybest = 3.52cm + 2.35cm = 5.87cm ∆q = (∆x) 2 + (∆y ) 2 = (0.05)2 + (0.04) 2 = 0.06cm q = 5.87cm ± 0.06cm Multiplying or Dividing Measurements with Uncertainties Suppose you make two measurements, x = xbest ± ∆x y = ybest ± ∆y What is the uncertainty in the quantity q = xy or q = x/y? To obtain the uncertainty we will find the highest and lowest probable value of q = xy. The result will be the same if we consider q = x/y. Again we would like to state q in the standard form of q = qbest ± ∆q where now qbest = xbest ybest. (highest probable value of q = xy): (xbest+ ∆x)(ybest + ∆y) = xbestybest + xbest ∆y +∆x ybest + ∆x ∆y = qbest + ∆q = xbestybest + (xbest ∆y + ∆x ybest) = qbest + ∆q (lowest probable value of q = xy): 2
3. 3. (xbest- ∆x)(ybest - ∆y) = xbestybest - xbest ∆y - ∆x ybest + ∆x ∆y = qbest – ∆q = xbestybest – (xbest ∆y + ∆x ybest) = qbest – ∆q Since the uncertainties ∆x and ∆y are assumed to be small, then the product ∆x ∆y ≈ 0. Thus, we see that ∆q = xbest ∆y +∆x ybest in either case. Dividing by xbestybest gives ∆q x ∆y y ∆x = best + best xbest y best xbest y best xbest y best ∆q ∆y ∆x = + qbest y best x best Again, a similar result applies if we needed to obtain the uncertainty in the division of q = x/y. If we had multiplied or divided more than two measurements x, y, ......, z each with its own uncertainty ∆x, ∆y, ......... , ∆z respectively, the result would be ∆q ∆y ∆x ∆z = + + ........ + qbest y best x best z best Now, if the uncertainties ∆x, ∆y, ........., ∆z are random and independent, the result is 2 2 2 ∆q  ∆y   ∆x   ∆z  =  y  +  x  + ........ +   z   qbest  best   best   best  Ex. x = 49.52cm ± 0.08cm y = 189.53cm ± 0.05cm Calculate q = xy We would like to state q in the standard form of q = qbest ± ∆q xbest = 49.52cm, ∆x = 0.08cm ybest = 189.53cm, ∆y = 0.05cm qbest = xbestybest = (49.52cm)(189.53cm)=9.38553 x 103 cm2 2 2 2 ∆q  ∆x   ∆y   0.08cm   0.05cm   =   +  y   =  49.52cm  +  189.53cm  = 1.63691E − 3 qbest  xbest   best      3
4. 4. ∆q = (1.63691 x 10-3)qbest = (1.63691 x 10-3) (9.38553 x 103 cm2) ∆q = 15.3632cm2 ≈ 20 cm2 q = 9390 cm2 ± 20 cm2 Uncertainty for a Quantity Raised to a Power If a measurement x has uncertainty ∆x, then the uncertainty in q = xn, is given by the expression ∆q ∆x = n qbest xbest Ex. Let q = x3 where x = 5.75cm ± 0.08 cm. Calculate the uncertainty ∆q in the quantity q. We would like to state q in the standard form of q = qbest ± ∆q n=3 ∆x = 0.08cm xbest = 5.75cm qbest = xbest = 190.1cm 3 3 ∆q ∆x = n = qbest xbest ∆q  0.08cm  3 = (3)  190.1cm  5.75cm  ∆q = 7.93cm 3 ≈ 7cm 3 q = 190cm 3 ± 7cm 3 4