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Estimating Uncertainties InExperimental Results All experimental scientists need to know how well they can trust their results. The results of any experiment are only as valid as the degree of error in those results. A lot of time, effort , and money has been spent by scientists developing more “accurate” machines to measure events more precisely. This unit is all about making and keeping tracks of errors during experimental measurements.
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Estimating Uncertainties InExperimental ResultsExamine the image show below: What is the diameter of the tennis ball in cm? (answer: ~ 6.4 cm)
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Estimating Uncertainties InExperimental Results Does this mean it is exactly 6.4 cm? Could the diameter be 6.3 or 6.5 or even 6.44 cm? Look again…
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Estimating Uncertainties InExperimental Results All measured values must be accompanied by an estimate of the error or uncertainty associated with the measured value. The tennis ball has a diameter of 6.4 + 0.1 cm. Measurement value Estimated error value
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Estimating Uncertainties InExperimental Results Let’s look at some other possible ways of trying to report this value: 6.4 + 0.15 cm What is inconsistent here? 6 + 0.1 cm What is inconsistent here? 6.42 + 1 cm What is inconsistent here?
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Estimating Uncertainties InExperimental Results So what does 6.4 + 0.1 cm really mean? The real or actual diameter of the tennis ball lies between a maximum and a minimum value. The actual value lies Maximum value: 6.5 cm somewhere in between these two values! Minimum value: 6.3 cm We can not be any more precise than this!
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Estimating Uncertainties InExperimental ResultsTypes of Errors:Measurement errors fall into two main types: Systematic errors: These errors consistently influence a set of measurements in a particular direction , either too high or too low. These errors are associated with the precision of the measuring device (eg. not calibrated correctly), or errors in experimental procedures.
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Estimating Uncertainties InExperimental Results Random errors: These errors arise due to fluctuations in the experimental conditions or in the judgment of the experimenter. These errors are random, some being too high while others being too low and tend to average out if the experimenter repeats the experiment often enough.After you have identified the factors that may influence yourresults in the collection of experimental results, it is important todesign strategies to minimize both of these two types of errors.
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Estimating Uncertainties InExperimental ResultsThink: Drop a tennis ball from some height allowing it to hit the ground and measure the height to which it rebounds to. 1) Think and discuss all of the factors that could affect the outcome. 2) Think and discuss all of the possible error sources including both Systematic and Random.
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Estimating Uncertainties InExperimental Results Dealing with errors: Adding and Subtracting Measured Values:A student measures the mass of a 123.4 + 0.1 gbeaker + copper to be : A student measures the mass of a beaker to be : 113.8 + 0.1 g Mass of Copper is: 9.6 + ? g But what about the uncertainty? What happens to it? Does it stay at 0.1? Or does change to a higher or lower number?
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Estimating Uncertainties InExperimental ResultsThe rule is: When adding or subtracting numbers the numerical uncertainty is simply added! In order to determine the mass of copper the student subtracted two measured values: therefore simply add the numerical error! Mass of Copper is: 9.6 + 0.2 g Numerical error
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Estimating Uncertainties InExperimental Results Now try these:4.5 + 0.2 m + 2.3 + 0.1m + 6.3 + 0.3 m = 13.1 + 0.6 m 67.9 + 0.2 g - 45.7 + 0.2 g = 22.2 + 0.4 g (34.5 + 0.2 cm) + (12.3 + 0.3 cm) - (14.3 + 0.2 cm) = 32.5 + 0.7 cm (1.5 + 0.5 m) - (4.3 + 0.5 m) + (8.8 + 0.3 m) = 6+1m
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Estimating Uncertainties InExperimental ResultsMultiplying or Dividing Measured Values: This becomes a little more complicated.The rule is:When measured values are multiplied or divided the percentageerrors are added. What is a percentage error? Answer: a numerical error changed to be represented as a percentage of the measured value
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Estimating Uncertainties InExperimental ResultsHow is this done? Easily: Remember the copper: Mass of Copper is: 9.6 + 0.2 g 0.2 Percent error = X 100 = 2% 9.6 Mass of Copper is: 9.6 + 2 % g
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Estimating Uncertainties InExperimental ResultsFormula for finding Percentage Error: Numerical Error Percentage error = X 100 Measured Value
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Estimating Uncertainties InExperimental Results Now try These: Change numerical to percentage error: 13.1 + 0.6 m 13.1 + 5 % m 22.2 + 0.4 g 22.2 + 2 % g 32.5 + 0.7 cm 32.5 + 2 % cm 6+1m 6 + 17 % m
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Estimating Uncertainties InExperimental ResultsNow try these: Remember when measured values are multiplied or divided, add the percentage errors! 1) 22.2 cm + 2 % x 45.2 cm + 5% = 1000 cm2 + 7 % 2) 2.31 g + 2 % ÷ 0.76 mL + 3% = 3.0 g/mL + 5 % 3) 45 + 1 m x 342 + 3 m = 15400 m2 + 3 % 4) {(2.2 cm + 2 % x 5.4 cm + 5%) + 14 + 0.3 cm2} = Careful on this last one! 26 + 1 cm2
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Estimating Uncertainties InExperimental ResultsHow to determine the numerical error? 1) Reading a scale: • Use ½ of the smallest division 2) Fluctuating scale: • Look at the range of fluctuations and divide by 2 • 1/2(maximum value – minimum error)
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Estimating Uncertainties InExperimental ResultsGraduated Cylinder Volume = 12.3 + 0.3 mL 13 12 11 10 Fill water up to this point
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