Unformatted text preview: Random Error: A Critical Assessment
• Different people reading the scale [shown in the figure below] report a range of values representing their subjective interpolation between the markings. between • If you were to measure the pH of blood in your body, you would probably get different answers for blood from different parts of the body, and the pH at a given location probably would vary with time. would 1 Precision vs. Accuracy Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Company, 2 Reducing Error
Measurements of quality control samples allow (i) to Measurements recognize systematic errors and (ii) to estimate precision. recognize Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 3 Reducing Errors: Standard Reference Materials Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Company, 4 Reducing Errors: How can you recognize a systematic error if How you do not have a standard sample available? available? 5 Reducing Errors: How can you recognize a systematic error if How you do not have a standard sample available? available? 6 Reducing Errors: How can you recognize a systematic error if How you do not have a standard sample available? available? 7 The Simplest Way to Keep Track of Error: The Significant Figures Significant
How many significant figures does the How number 92,504 have? number 8 The Simplest Way to Keep Track of Error: The Significant Figures Significant
How many significant figures does the How number 92,500 have? number 9 How old is the Grand Canyon? 10 Significant Figures: Addition and Subtraction
The last significant digit is the same as that The for the least certain value. for Formula Weight of Ag2MoO4: 107.870 107.870 95.94 15.9994 15.9994 15.9994 15.9994 375.6776
11 (Ag) (Ag) (Mo) (O) (O) (O) (O) Why do we only have four digits for Mo but six for O? 107.870 107.870 95.94 15.9994 15.9994 15.9994 15.9994 375.6776 (Ag) (Ag) (Mo) (O) (O) (O) (O)
12 Rounding Rules
• Round to the nearest significant digit. Round • Only round the final answer. Only • If the digit to be rounded is exactly 5 then round to the closest even number. round 13 Multiplication and Division
The answer will have the same number of The significant digits as the least certain value. significant 35.63 x 0.5481 x 0.05300 1.1689 x 100% = 88.5470578% 14 Logarithms
The number of digits in the mantissa should The equal the number of significant figures in the original value. original log (1294) = 3.1119 15 Antilogarithms
The number of significant figures in the The antilogarithm should equal the number of digits in the mantissa. in Antilog (1.519) = 101.519 = 33.0 16 A Better Way: Propagation of Errors
Addition and Subtraction
2 2 e = e12 + e2 + e3 Example: During a titration the initial reading on a 50 mL Example: buret was 37.4 mL. The volume remaining in the buret after reaching the endpoint was 15.1 mL. What is the error in the volume dispensed? volume Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Company, 17 Propagation of Errors
Multiplication and Division erel = (e ) + (e ) + (e )
2 1 rel 2 2 rel 2 2 2 3 rel 2 εT = x T ε1 ε2 ε3 + + x x x 1 2 3 Example: 0.1053 g of NaOH is weighed on an analytical Example: balance, transferred to a 100 mL volumetric flask, and diluted to the mark. What is the error in the concentration of this solution? Assume the error in the mass measurement is ±0.0005 g. ±0.0005 18 Propagation of Errors
Mixed Operations • Perform propagation of error calculation for each operation in order. each Example: The NaOH solution prepared in the Example: example was used in the titration described earlier. the concentration of acid if titrating against HCl? the previous What is 19 Meaningful Confidence Intervals
1.022 ± 0.0114294 1.0223405 ± 0.011 1.0223405 ± 0.0114294 1.022 ± 0.011 1.0223 ± 0.0114
20 Sig. Figs. vs. Propagation of Error
• Propagation of error is a more rigorous treatment and will give a more accurate representation of the error. error. • The significant figures approach is useful when we need a quick estimate of the error. need * Significant figures approach: 0.821 ± 0.002 0.803 ± 0.002 = 1.022 ± 0.004 * For this example: What is the error of the result as For estimated by the “propagation of error” method? 21 estimated ...
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 Spring '10
 buhlman
 pH, Accuracy and precision, Observational error, Significance arithmetic, Daniel C. Harris

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