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Uncertainty in Measurement Professor Bob Kaplan University Department of Science 1

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Limitations of the Instrument Individual Skill Random Conditions ( not under control ) 2

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Numbers reported should have: 1) All Digits Known 2) One Estimated Digit 3

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All known digits and Estimated digit ( uncertain digit ) 4

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Smallest unit of measurement on glassware, ruler, scale, etc. Place value of increment or unit in the reported number. 8

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Degree of uncertainty generally depends on the smallest division or increment of the measuring instrument used (e.g. ruler, graduated cylinder, etc.). But it also depends on: Skill of the individual One person may feel comfortable splitting the Division in half ( +/- 0.5 unit ) Another person may feel confident splitting the Division in tenths ( +/- 0.1 unit ) 9

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Reported number: 34.746 meters Uncertainty level: +/- 0.001 meters = +/-- 1 mm Reported number: 34.73579 meters Uncertainty level: +/- 0.00001 meters = +/-- 0.01 mm 11

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In measured numbers, the “sig figs” include: All the reported numbers including the estimated digit. When we do calculations, we will need to count the significant figures in each of the numbers used. In each individual number, all non-zero numbers are counted as “sig figs”. Zeros may or may not be significant, depending on their position in the number. 12

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Zeros between integers Always significant [ e.g. 1004 ] 13

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Zeros that precede integers in decimals Never significant [ e.g. 0.0001234 ] 14

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Zeros that follow integers - End of a number Never significant [ e.g. 1,004,000 ] 15

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The purpose of significant figures is to tell you where to round off the number. If the first digit to be dropped is 4 or less, it and all the following digits should be dropped. If the first digit to be dropped is 5 or greater, the last retained digit of the number is increased by 1. 16

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Answer can be no more precise than the least precise quantity. Example: Solution of nitric acid that is precise to +/-0.0001 molarity (moles / liter). Mix that with another solution that was not measured precisely at all. Is the precision of my original solution retained ? Of course not !!!! 17

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Result should contain the same number of sig figs as the measurement that has the least number of sig figs. 18

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3 * 1, 465, 876 = 4, 000,000 3 * 1.465876 = 4.0 32 * 550 = 17,600 = 18,000 32 * 560 = 18,920 = 19,000 32 * 568 = 18,176 = 18,000 32 * 575 = 18,400 = 18,000 32 * 580 = 18,560 = 19,000 19

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“Limiting term” : Term with fewest decimal places. The result is rounded off the same as number with fewest decimal places. 20

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57898.32 + 33.34567 _____________ 57931.66 57931.67 21

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Consider the number: 564.32 What is the place value of the 5 ? Hundreds is correct In a number like 564.32, what single thing determines the uncertainty ??? Last number (or digit) is correct !! 22

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What about the last digit is important ??? Place Value ! What is the place value of the last digit in the number 546.32 ? Hundredths So what is the level of uncertainty ???? +/-.01 23

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Precision : Measures repeatability Accuracy : Distance from true value 24

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True value: 32.146 Accurate measurements: 32.132, 32.150, 32.161 Precise measurements: 36.456, 36.468, 36.345 25

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Systematic Error Instrumentation Calibration Standards of Measurement 26

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If you have come here directly from the SC155 Seminar session, please return to the KU course platform now to continue with the live session of discussion, questions and answers See you all there ! 27

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