Propagation of Uncertainty Copyrighted by T Darrel Westbrook

Propagation of Uncertainty Copyrighted © by T. Darrel Westbrook

Propagation of Uncertainty This presentation is derived from Appendix B of Physics for Scientists and Engineers, Volume 1, 6 th Edition, ISBN 0 -495 -27756 -8 Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 2

Propagation of Uncertainty There are two parts to dealing with uncertainty in science. The second part is covered in this presentation, Propagation of Uncertainty, and the first part covers data/number accuracy and precision in the presentation Significant Digits It is recommended you view Propagation of Uncertainty presentation second. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 3

Propagation of Uncertainty What You Will Learn: – About Uncertainty (Error) Propagation – How to determine confidence level of calculations when using approximate data (or numbers) Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 4

Propagation of Uncertainty There is always error or uncertainty with any measurement. Common errors are from the procedures we use to take measurements (method error), errors in the equipment we use (instrumentation error), physical limitations (parallax error), etc. To understand the limitations of our measurements we have to have confidence in the accuracy of our data and calculations. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 5

Propagation of Uncertainty If we measure a CD label with a ruler that is readable to two significant digits and our accuracy is 0. 1 cm then the data can be expressed as absolute uncertainty. Say (5. 5 0. 1) cm Other situations, will require us to use fractional uncertainty (or another form of fractional uncertainty, percent uncertainty). This allows us to combine the data errors to give us a confidence level for the final results. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 6

Propagation of Uncertainty Back to our CD label example, in absolute uncertainty (5. 5 0. 1) cm In fractional uncertainty it would be 5. 5 cm 0. 1/5. 5 = 5. 5 cm 0. 018 Or, as a percent uncertainty it is 5. 5 cm 1. 8% Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 7

Propagation of Uncertainty When we use various data in calculations (including its built-in uncertainty or error), the error gets bigger with each calculation. This is called propagation of uncertainty or error propagation. The more complicated the calculations the more error. The rules that follow permit us to determine the combination of those errors and provides a reasonable estimate of the overall error. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 8

Propagation of Uncertainty Multiplication (and Division) - add the percent uncertainties. (5. 5 cm 1. 8%) (6. 4 cm 1. 6%) = (35. 2 cm 3. 4%) Only two significant digits 35 cm 3. 4% Return back to absolute uncertainty (35 1. 19) cm Only one significant digit in the uncertainty (35 1) cm Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 9

Propagation of Uncertainty Addition (and Subtraction) - add the absolute uncertainties. Note the difference. Multiplication (and Division) - add the percent uncertainties. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 10

Propagation of Uncertainty Addition (and Subtraction) - add the absolute uncertainties. (5. 5 0. 1) cm + (6. 4 0. 1) cm = (11. 9 0. 2) cm Only two significant digits (12 0. 2) cm Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 11

Propagation of Uncertainty Powers - multiply the percent uncertainties by the power. (5. 5 cm 1. 8%)3 cm = 166. 375 cm 5. 4% Only two significant digits 170 cm 5. 4% Return back to absolute uncertainty (170 9. 18) cm Only one significant digit in the uncertainty (170 9) cm Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 12

Propagation of Uncertainty Notice that the uncertainties are all added. So, if you have enough operations and the resulting number is not very large, you could get a cumulated error that is larger than the answer. Especially, for calculations that require subtraction where the numbers are close together. Suppose; (5. 5 0. 2) cm – (5. 0 0. 2) cm = (0. 5 0. 4) cm It is easy to see, that the error propagation is quite large compared to the answer (an 80% error). Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 13

Propagation of Uncertainty What You Have Learned: – About Uncertainty (Error) Propagation – How to determine confidence level of calculations when using approximate data (or numbers) Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 14

Propagation of Uncertainty End of Line Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel

Propagation of Uncertainty Our confidence level in our data or numbers is expressed in the same units as the data being measured or the number used. That confidence level is called absolute uncertainty. An example of absolute uncertainty in the presentation is our measure of a CD label of (5. 5 0. 1) cm. In secondary math classes that absolute uncertainty is stated as 5. 4 x 5. 6 or using absolute value, it is | x – 5. 5 | 0. 1. Two other forms of absolute uncertainty is fractional uncertainty and percent uncertainty. Fractional Uncertainty = absolute uncertainty/measure = 0. 1/ 5. 5 = 0. 018 Percent Uncertainty is just fraction uncertainty in the form of a percent 0. 018 = 1. 8% Error Propagation (Propagation of Uncertainty) – is a result of calculations. The error get bigger with each successive operation and it must be accounted fro in order to determine the confidence level is the final answer of the calculations. Rules Addition (and Subtraction) - add the absolute uncertainties. Note the difference. Multiplication (and Division) - add the percent uncertainties. Powers - multiply the percent uncertainties by the power. Propagation. Of. Uncertainty. ppt Copyrighted © T. Darrel Westbrook 16