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 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

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

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

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.

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

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

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

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!

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

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

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

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

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

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

Propagation of Uncertainties

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

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

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)

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

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

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

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

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

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