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Assigning and Propagating Uncertainties

Assigning and Propagating Uncertainties. Assigning Uncertainties. In the laboratory experiments you will be performing you will have to make measurements; the results of these measurements will always be uncertain to some extent

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Assigning and Propagating Uncertainties

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  1. Assigning and Propagating Uncertainties

  2. Assigning Uncertainties • In the laboratory experiments you will be performing you will have to make measurements; the results of these measurements will always be uncertain to some extent • For example, suppose you use a balance to determine that the mass (m) of some chemical constituent used in your experiment is 15.87g; but your “weighing” apparatus only had two decimal places and the mass could be anywhere from 15.86 to 15.88g • Your best estimate was 15.87g and there was uncertainty of 0.01g • You would then report your result as: m = 15.87g ± 0.01g NOTE: THE UNITS (grams) APPLY TO BOTH THE MEASUREMENT AND THE UNCERTAINTY

  3. Assigning Uncertainties YOU SHOULD ASSIGN AN UNCERTAINTY TO EVERY MEASUREMENT THAT YOU MAKE!!! • In the example on the previous slide, the uncertainty resulted from a limitation in the resolution of the instrument used; you might use a balance with a higher resolution (three decimal places) and find m = 15.868g ± 0.001g

  4. Assigning Uncertainties • For example, you may measure the time (t) that it takes for a reaction to occur • On successive tries you find t = 1.3, 1.4, 1.2, 1.5, 1.4, 1.4, 1.3, and 1.5 seconds • It might then be reasonable to report your results as: t = 1.4 ± 0.1 seconds, since the range from 1.3 to 1.5 seconds encompasses almost all of the experimental readings

  5. Assigning Uncertainties • No hard and fast rules are possible for assigning uncertainties, instead you must be guided by common sense. • If the space between the scale divisions is large, you may be comfortable in estimating to 1/5 or 1/10 of the least count. • If the scale divisions are closer together, you may only be able to estimate to the nearest 1/2 of the least count • If the scale divisions are very close you may only be able to estimate to the least count.

  6. Assigning Uncertainties • A solid rule of thumb when trying to determine an uncertainty for a given measuring instrument is: “Half the smallest unit measured” • For example, when using a graduated cylinder that has 1 mL gradations the uncertainty would be ±0.5mL • It is also helpful to look at the measurement instrument and see if an uncertainty is printed on the instrument

  7. Absolute and Relative Uncertainty • Uncertainty can be expressed in two different ways • Absolute uncertainty refers to the actual uncertainty in a quantity • For example, the have three mass measurements: 6.3302g 6.3301g 6.3303g • The average is 6.3302 ±0.0001g • The absolute uncertainty is 0.0001g

  8. Absolute and Relative Uncertainty • Relative uncertainty expresses the uncertainty as a fraction of the quantity of interest • Other ways of expressing relative uncertainty are in per cent, parts per thousand, and part per million • For the following example of an object with mass 6.3302 ±0.0001g, the relative uncertainty is 0.0001g / 6.3302g which is equal to 2 × 10-3 percent, or 2 parts in 100, 000 or 20 parts per million • Relative uncertainty is a good way to obtain a qualitative idea of the precision of your data and results

  9. Absolute and Relative Uncertainty • In general, results of observations should be reported in such a way that the last digit given is the only one whose value is uncertain due to random errors • Note that systematic and random errors refer to problems associated with making measurements. • Mistakes made in the calculations or in reading the instrument are not considered in error analysis. • It is assumed that the experimenters are careful and competent!

  10. Definition of Propagation of Uncertainties • Once you have assigned uncertainties to your experimentally measured quantities, you will probably have to combine these quantities in some way in order to determine some derived quantity • For example, you measure the distance that an object traveled, say, 1.00 ± 0.01 meters and the time during which it moved, say 2.0 ± 0.1 seconds; now you want to know the average speed of that object

  11. Definition of Propagation of Uncertainties • How will you combine the uncertainties in the two measured quantities in order to find the uncertainty in speed? • The manner in which uncertainties combine to give the resultant uncertainty is called propagation of uncertainties

  12. Propagation of Uncertainties • There are three different ways of calculating or estimating the uncertainty in calculated results

  13. Propagation of Uncertainties • Significant Figures: the easy way out • Use the implicit uncertainty in each measurement’s significant figures to determine significant figures in the result by following the rules for adding, subtracting, multiplying and dividing • Useful when a more extensive uncertainty analysis is not needed. A reasonable estimate of the uncertainty should always be implied by the significant figures in any calculated result

  14. Propagation of Uncertainties 2. Uncertainty Propagation: Not as bad as it looks • Uses uncertainty or precision of each measurement, arising from limitations of measuring devices. Contribution of each uncertainty to the final result is calculated • Useful for limited number or single measurements

  15. Propagation of Uncertainties 3. Statistical Methods: When you have lots of numbers • Uses the spread in the values of many repeated results to estimate the uncertainty in their average • Useful for many repeated measurements and for fits to analytical equations, linear or otherwise

  16. Propagation of Uncertainties

  17. Propagation of Uncertainties • SIGNIFICANT FIGURE ANALYSIS: Multiplication and division: • The results has the same number of significant figures as the smallest of the number of significant figures for any value used in the calculation Addition and subtraction: • The result will have a last significant digit in the same place as the left-most of the last significant digits of all the numbers used in calculation

  18. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Addition and Subtraction: • When adding or subtracting numbers indicating uncertainties you also add the uncertainties Example: • 1384 ± 2g + 111 ± 2 g = 1495 ± 4g • 45.34 ± 0.05ºC – 23.34 ± 0.05ºC = 22.0 ± 0.1ºC

  19. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Multiplication and Division: • When multiplying or dividing numbers indicating uncertainties you add the relative uncertainties (percent uncertainties) Example: • 2810 ± 4g ÷ 7.43 ± 0.05 dm3 • 2810 g ± 0.14% ÷ 7.43dm3 ± 0.67% = 378 g dm-3 ± 0.81% • 378 ± 3 g dm-3 (This is reasonable because the uncertainty matches the last place of significance)

  20. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Exponents and Square Roots: • Exponents are essentially multiplication so, for x2 the uncertainty is equal to 2 times the relative uncertainty (percent uncertainty) • 2(Δx/x) • For a cubed unit the uncertainty would be 3 times the relative uncertainty • A square root would be ½ times the uncertainty

  21. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Exponents and Square Roots: • Exponents are essentially multiplication so, for x2 the uncertainty is equal to 2 times the relative uncertainty (percent uncertainty) • 2(Δx/x) • For a cubed unit the uncertainty would be 3 times the relative uncertainty • A square root would be ½ times the uncertainty

  22. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Logarithms: • x ± a log x • the new uncertainty is calculated by: Xnew = 0.434 (a/x) ln x • the new uncertainty is calculated by: xnew = a/x

  23. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Pure Numbers: • This refers to numbers such as 2, ½, π, etc. • In general, a division of a measurement by 2 will reduce the absolute uncertainty by 2 Example: 43.2 cm ± 2.6 cm x 3.0 = = 129.6 cm ± 7.8 cm

  24. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Uncertainty In A Mean: • An average value will have the same number of significant figures as the least precise measured value • A reasonable idea of the uncertainty can be obtained by dividing the range of values by the number of values • Uncertainty in mean ~ xmax – xmin / n

  25. Propagation of Uncertainties 2. UNCERTAINTY PROPAGATION: Uncertainty In A Mean: Example: Five temperature measurements were taken: 23.1ºC, 22.5ºC, 21.9ºC, 22.8ºC, 22.5ºC Determine the mean and uncertainty for this data: Mean = 23.1 + 22.5 + 21.9 + 22.8 + 22.5 / 5 = 22.6ºC Uncertainty in mean ~ 23.1 – 21.9/5 = 0.2 So the mean temperature is 22.6 ± 0.2ºC

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