Chapter 6: Random Errors in Chemical Analysis

1. Chapter 6: Random Errors in Chemical Analysis

2. Contents in Chapter 06 1. The Nature of Random Errors 1) Random Errors and Their Mathematical Equations 2) Central Limit Theorem 2. Gausian (Normal) Distribution 1) Define Gausian Distribution 2) Z Value Transformation 3) Z value Applications 4) Constructing Gaussian Curve From Experimental Data 3. Standard Deviation of Calculated Results 4. Significant Figures

3. 1. The Nature of Random Errors 1) Random Errors and Their Mathematical Equations • Random (Indeterminate, Statistic) error: Errors affecting the precision (uncertainty), caused by uncontrollable variables. Positive and negative () fluctuation occur with approximate equal frequency. • Population: The set of infinite objects in the system being investigated. • Sample: The finite members of a population that we actually collect and analyze.

4. Spread (range, w): The difference between extreme values in a set of d • Mean (Average) • Standard deviation - For population: - For sample: ●

5. Pooled standard deviation: For several subsets of data, estimating standard deviation by pooling (combining) the data. • Variance = s2 *variance are additive • Relative standard deviation (RSD) by 100% (also called coefficient of variation, CV):

6. Degree of freedom (dof) In statistics, the number of independent observations on which a result is based. - For standard deviation: dof = n-1, n is munber of measurement. - For pooled standard deviation: dof = N – M, N: Total number of measurements M: Number of the subset.

7. Central Limit Theorem i) Definition • For the probability distribution plot for the frequency of individual values. The measurements subject to indeterminate errors arise a normal (Gaussian) distribution. • The sources of individual error must be independent. • The individual error must have similar magnitude (no one source of error dominates the final distribution).

8. ii) Simulated Central Limit Theorem (Frequency of Combinations of fourequal-sized uncertainty, u)

9. 4 random uncertainties

10. 10random uncertainties • Infinite number of random uncertainties - A Gaussian (normal) distribution curve. - Symmetrical about the mean.

11. 2. Gausian (Normal) Distribution • Define Gausian Distribution A “bell-shaped” probability distribution curve for measurements showing the effect of random error, which encountered continuous distribution. The equation for Gaussian distribution:

12. IF: Same Mean, Different SD 982_Ch04_Treat_RandDistr_Data.xls

13. IF: Different Mean, Same SD 982_Ch04_Treat_RandDistr_Data.xls

14. IF: Different Mean, Different SD 982_Ch04_Treat_RandDistr_Data.xls

15. 2) Z value Transformation • For x-axis: transform x to z by: • For y-axis: transform to f(z) by :

16. Now, the Gaussian Curve Always Consistent 982_Ch04_Treat_RandDistr_Data.xls

17. 教戰守則 • 經過 z transform, Gaussian curve 長得一模一樣，此時 x-axis 的單位為σ。 • 看到標示為 x，單位為 σ時，代表已經經過 z transform。 • Relative frequency (y) at mean (x), 約為 0.4。 • The entire area of the Gaussian curve is 1.

18. Z value Applications Area (probability, percentage) in defined z interval

19. Constructing Gaussian Curve From Experimental Data i) General procedure Step 1: Raw data collection Step 2: Arrange the data in order from lowest to highest Step 3: Condense the data by grouping them into cells Step 4: Pictorial representation of the frequency distributions Step 5: Estimating the σ from s Step 6: Plotting relative frequency versus x or z

20. Example: Replicate Data for the Calibration of a 10 mL Pipet 982_Ch04_Treat_RandDistr_Data.xls

21. Cont’d 982_Ch04_Treat_RandDistr_Data.xls

22. 982_Ch04_Treat_RandDistr_Data.xls

23. Number in range Percentage in range Relative frequency versus x 962_Ch04_Treat_RandDistr_Data.xls X-axis 刻度為體積之間隔

24. Matching Histogram with Gaussian curve

25. Z Transformed Gaussian Curve 982_Ch04_Treat_RandDistr_Data.xls

26. Standard Deviation of Calculated Results (for Random Errors)

27. 1) Sum or Difference Example 1: The calibration result of class A 10 mL pipet showed that the marker reading is 9.9920.006 mL. When it is used to deliver two successive volumes. What is the absolute and percent relative uncertainties for the total delivered volume. Solution: Total volume = 9.992 mL + 9.992 mL = 19.984 mL Ans: 19.984(0.008) mL

28. Example 2: For a titration experiment, the initial reading is 0.05(0.02) mL and final reading is 17.88 (0.02) mL. What is the volume delivered? Solution: Delivered volume = 17.88 mL – 0.05 mL = 17.83 mL Ans: 17.83(0.03) mL

29. 2) Product or Quotient Example 1: The quantity of charge Q = I x t. When a current of 0.150.01 A passes through the circuit for 1201 s, what is the total charge? Solution: Total charge 0.15 A x 120 s = 18 C Ans: 18(1) C

30. 3) Mixed operations

31. 4) Exponents and logarithms Example: The pH of a solution is 3.720.03, what is the [H3O+] of this solution? Solution: [H3O+] = 10–pH y = 10–3.72 =1.91x10–4 y(sy) = 1.91 (0.13)x10–4 Ans: 1.9 (0.1)x10–4 M

32. 4. Significant Figures • Statement of significant figures • The digits in a measured quantity, including all digits known exactly and one digit (the last figure) whose quantity is uncertain. • The more significant digits obtained, the better the precision of a measurement. • The concept of significant figures applies only to measurements. • The Exact values (e.g., 1 km = 1000 m) have an unlimited number of significant figures.

33. 2) Recording with significant figures 0 1 2 3 Recording: 1.4 or 1.5 or 1.6 2 significant figure, 1 certain, 1 uncertain 0 1 2 3 3 significant figure, 2 certain, 1 uncertain Recording: 1.51 or 1.52 or 1.53

34. 3) Rules for Zeros in Significant Figures • Zeros between two other significant digits are significant e.g. 10023 no. of sig. fig.: 5 • A zero preceding a decimal point is not significant e.g., 0.10023 no. of sig. fig.: 5 • Zeros preceding the first nonzero digit are not significant e.g. 0.0010023 no. of sig. fig.: 5 • Zeros at the end of a number are significant if they are to the right of the decimal point e.g. 0.1002300 no. of sig. fig.: 7

35. Zeros at the end of a number may or may not be significant if the number is written without a decimal point e.g. 92500 no. of sig. fig.: N/A Scientific notation is required: e.g. 9.25x104 no. of sig. fig.: 3 e.g. 9.250x104 no. of sig. fig.: 4 e.g. 9.2500x104 no. of sig. fig.: 5

36. Significant Figures in Arithmetic • Rules for rounding off numbers • If the digit immediately to the right of the last sig. fig. is more than 5, round up. • If the digit immediately to the right of the last sig. fig. is less than 5, round down. • If the digit immediately to the right of the last sig. fig. is 5 followed by nonzero digits, round up. • If the digit immediately to the right of the last sig. fig. is 5 • round up if the last sig. fig. is odd. • round down if the last sig. fig. is even. v) If the resulting number has ambiguous zeroes, it should be recorded in scientific notation to avoid ambiguity.

37. Examples: 35.76 in 3 sig. fig. is 35.8 35.74 in 3 sig. fig. is 35.7 24.258 in 3 sig. fig. is 24.3 24.35 in 3 sig. fig. is 24.4 (rounding to even digit) 24.25 in 3 sig. fig. is 24.2 (rounding to even digit) 13,052 in 3 sig. fig. is 1.31 x 104

38. Example 1: Example 2: MW of KrF2 49.146 m + 72.13 m – 9.1 m 112.176 m 18.9984032 (F) + 18.9984032 (F) + 83.798 (Kr) 121.7948064 Ans: 112.2 m Ans: 121.795 g/mol 2) Addition and Subtraction: • The reported results should have the same number of decimal places as the number with the fewest decimal places

39. Note: To avoid accumulating “round-off ” errors in calculations: • Go through all calculation by calculator, then rounding on the final answer OR 2) Retaining one extra insignificant figure (a subscribe digit) for intermediate results, then rounding on the final answer

40. 2.432 x 106 + 0.06512 x 106 – 0.1227 x 106 2.37442 x 106 Ans: 2.374x106 Example 3: Write the answer with correct number of digits: 2.432x106 + 6.512x104– 1.227x105 = ?

41. Multiplication and Division: • The reported results should have no more significant figures than the factor with the fewest significant figures Example 1: 1.827 m × 0.762 m = ? 1.827 m x 0.762 m = 1.392174 m2 = 1.39 m2 Ans: 1.39 m2 Example 2: (4.3179 x 1012)(3.6x10–19) = 1.554444x10–6 = 1.6x10–6

42. Stoichiometric coefficients in a chemical formula, and unit conversion factors etc, have an infinite number of significant figures. Example: Results of four measurements: 36.4 g, 36.8 g, 36.0 g, 37.1 g. What is the average? Solution: (36.4+36.8+36.0+37.1)/4 = 36.6 Ans: 36.6 g

43. 4) Logarithms and Antilogarithms • Mathematic terms • For exponential expression, e.g. 4.2 x 10–2: “4.2” is called the coefficient, “–2” is called the exponent. • For logarithm operation, e.g. log (4.2 x 10–2) = –1.38 “–1” is called the characteristic, the integer part “38” is called the mantissa, the decimal part

44. ii) Logarithms operation Example 1: log(56.7 x 106) = ? Solution: log(5.67 x 107) = 7 + log(5.67) = 7.754 3 sig. fig. Mantissa remain equal number of sig. fig. as the original digit in coefficient Example 2: log(0.002735) = ? Solution: log(0.002735) = – 2.5630 4 sig. fig. Or log(0.002735) = 2.735x10–3 = (– 3) + log(2.735) = (–3) + (0.4370) = – 2.5630

45. Example 3: Percent transmittance (%T) is related to the absorbance (A) by the equation:  A = –log (%T/100). What is correct digits of A if %T is 72.9. Solution: A = –log (%T/100) = –log (0.729) = – (– 0.137) = 0.137 Or log (0.729) = – log(7.29x10–1) = – [(–1) + log (7.29)] = – [(–1) + (0.863)] = 0.137 Example 4: The pH is defined as pH = –log [H3O+]. If the [H3O+] is 3.8 x 10–2 M, what is the pH of the solution? Solution: pH = –log [H3O+] = –log (3.8 x 10–2) = –[(–2) + log(3.8)] =– (–1.42) = 1.42

46. iii) Antilogarithms operation Example 1: antilog(2.671) = ? Solution: antilog(2.671) = 102.671 = 100.671x102 = 4.69 x 102 Example 2: What is the correct digits of %T if the absorbance is 0.931? Solution: Antilog (–0.931) = 10–0.931 = 0.117 %T = 0.117 x 100% = 11.7% Final digit has same number of sig. fig. as number of digits in mantissa

47. Example 3: If the pH is 10.3, what is the [H3O+]? Solution: [H3O+] = 10–pH = 10–10.3 = 5 x 10–11 Or 10–10.3 = 100.7 x 10–11 = 5 x 10–11 Example 4: If the pH is 2.52, what is the [H3O+]? Solution: [H3O+] = 10–pH = 10–2.52 = 3.0 x 10–3 Or 10–2.52 = 100.48 x 10–3 = 3.0 x 10–3

48. Homework (Due 2014/3/6) Skoog 9th edition, Chapter 06 Questions and Problems 6-1 6-4 6-5 6-9 (a) (c) (e) 6-11 (a) (c) 6-15 End of Chapter 06