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  1. Chapters 3Uncertainty January 30, 2007 Lec_3

  2. Outline • Homework Chapter 1 • Chapter 3 • Experimental Error “keeping track of uncertainty” • Start Chapter 4 • Statistics

  3. Homework Chapter 1 – “Solutions and Dilutions” Questions: 15, 16, 19, 20, 29, 31, 34

  4. Chapter 3 Experimental Error And propagation of uncertainty

  5. Keeping track of uncertainty Significant Figures Propagation of Error 35.21 ml 35.21 (+ 0.04) ml

  6. Suppose You determine the density of some mineral by measuring its mass • 4.635 +0.002 g And then measured its volume • 1.13 + 0.05 ml

  7. Significant Figures (cont’d) • The last measured digit always has some uncertainty.

  8. 3-1 Significant Figures • What is meant by significant figures? Significant figures:

  9. Examples • How many sig. figs in: • 3.0130 meters • 6.8 days • 0.00104 pounds • 350 miles • 9 students

  10. “Rules” • All non-zero digits are significant • Zeros: • Leading Zeros are not significant • Captive Zeros are significant • Trailing Zeros are significant • Exact numbers have no uncertainty (e.g. counting numbers)

  11. Reading a “scale”

  12. What is the “value”? When reading the scale of any apparatus, try to estimate to the nearest tenth of a division.

  13. 3-2Significant Figures in Arithmetic • We often need to estimate the uncertainty of a result that has been computed from two or more experimental data, each of which has a known sample uncertainty. Significant figures can provide a marginally good way to express uncertainty!

  14. 3-2Significant Figures in Arithmetic • Summations: • When performing addition and subtraction report the answer to the same number of decimal places as the term with the fewestdecimal places +10.001 + 5.32 + 6.130 ?

  15. Try this one 1.632 x 105 4.107 x 103 0.984 x 106 0.1632 x 106 0.004107 x 106 0.984 x 106 + +

  16. 3-2Significant Figures in Arithmetic • Multiplication/Division: • When performing multiplication or division report the answer to the same number of sig figs as the least precise term in the operation 16.315 x 0.031 = ? 0.505765 0.51

  17. 3-2Logarithms and Antilogarithms From math class: log(100) = 2 Or log(102) = 2 But what about significant figures?

  18. 3-2Logarithms and Antilogarithms Let’s consider the following: An operation requires that you take the log of 0.0000339. What is the log of this number? log (3.39 x 10-5) = log (3.39 x 10-5) = log (3.39 x 10-5) =

  19. 3-2Logarithms and Antilogarithms • Try the following: Antilog 4.37 =

  20. “Rules” • Logarithms and antilogs 1. In a logarithm, keep as many digits to the right of the decimal point as there are sig figs in the original number. 2. In an anti-log, keep as many digits are there are digits to the right of the decimal point in the original number.

  21. 3-4. Types of error • Error – difference between your answer and the ‘true’ one. Generally, all errors are of one of three types. • Systematic (aka determinate) – problem with the method, all errors are of the same magnitude and direction (affect accuracy) • Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision) • Gross. – occur only occasionally, and are often large.

  22. Absolute and Relative Uncertainty • Absolute uncertainty expresses the margin of uncertainty associated with a measurement. Consider a calibrated buret which has an uncertainty + 0.02 ml. Then, we say that the absolute uncertainty is + 0.02 ml

  23. Absolute and Relative Uncertainty • Relative uncertainty compares the size of the absolute uncertainty with its associated measurement. Consider a calibrated buret which has an uncertainty is + 0.02 ml. Find the relative uncertainty is 12.35 + 0.02, we say that the relative uncertainty is

  24. 3-5. Estimating Random Error (absolute uncertainty) • Consider the summation: + 0.50 (+ 0.02) +4.10 (+ 0.03) -1.97 (+ 0.05) 2.63 (+ ?)

  25. 3-5. Estimating Random Error • Consider the following operation:

  26. Try this one

  27. 3-5. Estimating Random Error • For exponents

  28. 3-5. Estimating Random Error • Logarithms antilogs

  29. Question • Calculate the absolute standard deviation for a the pH of a solutions whose hydronium ion concentration is 2.00 (+ 0.02) x 10-4

  30. Question • Calculate the absolute value for the hydronium ion concentration for a solution that has a pH of 7.02 (+ 0.02) [H+] = 0.954992 (+ ?) x 10-7

  31. Suppose You determine the density of some mineral by measuring its mass • 4.635 +0.002 g And then measured its volume • 1.13 + 0.05 ml What is its uncertainty? =4.1 +0.2 g/ml

  32. The minute paper Please answer each question in 1 or 2 sentences • What was the most useful or meaningful thing you learned during this session? • What question(s) remain uppermost in your mind as we end this session?

  33. Chapter 4 Statistics

  34. General Statistics Principles • Descriptive Statistics • Used to describe a data set. • Inductive Statistics • The use of descriptive statistics to accept or reject your hypothesis, or to make a statement or prediction • Descriptive statistics are commonly reported but BOTH are needed to interpret results.

  35. Error and Uncertainty • Error – difference between your answer and the ‘true’ one. Generally, all errors are of one of three types. • Systematic (aka determinate) – problem with the method, all errors are of the same magnitude and direction (affect accuracy). • Random – (aka indeterminate) causes data to be scattered more or less symmetrically around a mean value. (affect precision) • Gross. – occur only occasionally, and are often large. Can be treated statistically.

  36. The Nature of Random Errors • Random errors arise when a system of measurement is extended to its maximum sensitivity. • Caused by many uncontrollable variablesthat are an are an inevitable part of every physical or chemical measurement. • Many contributors – none can be positively identified or measured because most are so small that they cannot be measured.

  37. Random Error • Precision describes the closeness of data obtained in exactly the same way. • Standard deviation is usually used to describe precision

  38. Standard Deviation • Sample Standard deviation (for use with small samples n< ~25) • Population Standard deviation (for use with samples n > 25) • U = population mean • IN the absence of systematic error, the population mean approaches the true value for the measured quantity.

  39. Example • The following results were obtained in the replicate analysis of a blood sample for its lead content: 0.752, 0.756, 0.752, 0.760 ppm lead. Calculate the mean and standard deviation for the data set.

  40. Standard deviation • 0.752, 0.756, 0.752, 0.760 ppm lead.

  41. Distributions of Experimental Data • We find that the distribution of replicate data from most quantitative analytical measurements approaches a Gaussian curve. • Example – Consider the calibration of a pipet.

  42. Replicate data on the calibration of a 10-ml pipet.

  43. Frequency distribution

  44. The minute paper Please answer each question in 1 or 2 sentences • What was the most useful or meaningful thing you learned during this session? • What question(s) remain uppermost in your mind as we end this session?