Statistics lectures
- 1. Experimental Error Every Physical Measurement has Error • Preparation of a Solution • Measurement of pH No way to measure the “TRUE” value of anything
- 2. Experimental Error Every Physical Measurement has Error • Preparation of a Solution • Measurement of pH No way to measure the “TRUE” value of anything Repetition: evaluate the reproducibility (Precision) Different Methods: Confidence near the “truth” (Accuracy)
- 3. Experimental Error Every Physical Measurement has Error • Preparation of a Solution • Measurement of pH No way to measure the “TRUE” value of anything Repetition: evaluate the reproducibility (Precision) Different Methods: Confidence near the “truth” (Accuracy) • Precision – describes the reproducibility of results – involves a comparison btwn measurements • Accuracy – nearness of a measurement to its accepted value
- 4. Types of Error Types of Error 1. Determinate (systematic) 2. Indeterminate (random)
- 5. Types of Error 1. Systematic (Determinate) Error Definite value; Assignable Cause In principle measured and accounted Unidirectional low or high
- 6. Types of Error 1. Systematic (Determinate) Error 2. Random (Indeterminate) Error
- 7. Types of Error 1. Systematic (Determinate) Error 2. Random (Indeterminate) Error – Natural Limitations – Uncontrollable – Random – Ultimate Limitation – Use Statistics
- 8. Detect Known samples Blank samples Different people/laboratories Different Methods
- 9. Types of Error Clicker Question Which of the following represents a systematic error a. Using a pH meter that has not been calibrated b. Missing the end point in a titration due to color blindness c. Missing the end point in a titration due to slow reactions d. All of the above e. None of the above
- 10. Evaluate Uncertainty in the Result Significant Figures Absolute / Relative Uncertainty Propagation of Error Statistics Accept/Reject Data Point (Q-test) Define interval around a mean with given probability Types of Error
- 11. Clicker question How many significant figures does 0.00110 have a. 5 b. 4 c. 3 d. 2
- 12. Absolute / Relative Uncertainty Absolute margin of uncertainty associated with a measurement
- 13. Absolute / Relative Uncertainty/Error Absolute margin of uncertainty associated with a measurement ex. Tolerance on volumetric flask, pipet, buret,... 100.00 mL (±0.08) mL ex. Standard deviation ex. Xi – Xt (where Xt is true value)
- 14. Relative absolute uncertainty / magnitude of measurement think about it as being “relative” to something Xi – Xt (where Xt is true value) Xt Relative Standard Deviation S ___ X x 100%
- 15. Clicker questions If the average of three repetitive measurements is 3.05 grams and the standard deviation is 0.06, what is the relative percent standard deviation a. 2.0% b. 0.06% c. 6.0% d. 3.05% A student uses a gravimetric technique to determine how much nickel is in an unknown. She obtains a value of 3.06 %. The “true” value determined by the manufacturer through many trials is 3.03%. What is the percent relative error in the students data? a. 0.03% b. 0.09% c. 1% d. 3.03%
- 16. Propagation of Error • Used to deal with Random Error • Takes into account error assoc. with all individual measurements • Error of Computed Results – What is the uncertainty associated with the final result?
- 18. Propagation of Error Used to deal with Random Error Takes into account error assoc. with all individual measurements Error of Computed Results What is the uncertainty associate with the Final Result? Addition-Subtraction y = a + b + c Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c) Ey = (Ea 2 + Eb 2 + Ec 2 )1/2 Memorize Absolute Error
- 19. Propagation of Error Used to deal with Random Error Takes into account error assoc. with all individual measurements Error of Computed Results What is the uncertainty associate with the Final Result? Addition-Subtraction y = a + b + c Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c) Ey = (Ea 2 + Eb 2 + Ec 2 )1/2 Example: 0.05 (±0.02) + 4.10 (±0.03) - 1.97 (±0.05) Ey = ((0.02)2 + (0.03)2 + (0.05)2 )1/2 Ey = 0.062 Memorize Absolute Error
- 20. Propagation of Error Used to deal with Random Error Takes into account error assoc. with all individual measurements Error of Computed Results What is the uncertainty associate with the Final Result? Addition-Subtraction y = a + b + c Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c) Ey = (Ea 2 + Eb 2 + Ec 2 )1/2 Example: 0.05 (±0.02) + 4.10 (±0.03) - 1.97 (±0.05) Ey = ((0.02)2 + (0.03)2 + (0.05)2 )1/2 Ey = 0.062 Answer: 2.63 (±0.06) NOT: 2.63 (± 0.0620)
- 21. Propagation of Error 1. Addition and Subtraction 2. Division and Multiplication y = (a * b) / c Ea, Eb, Ec : uncertainty associated with indiv. #s (a, b,c) (Ea)r, (Eb)r, (Ec)r : relative uncertainties ex. (Ea)r = Sa / a (Ey )r= ((Ea)r 2 + (Eb)r 2 + (Ec)r 2 )1/2 Ey = (Ey)r * y Memorize Absolute Error Relative Error
- 22. Example Problem Calculate the Molarity and its uncertainty of a sodium hydroxide solution prepared by dissolving 0.0210 (±0.0002) g in a 100.00 (± 0.02) mL volumetric flask?
- 23. Statistical Treatment of Random Errors Evaluate random error Assumption: random errors follow a gaussian (normal) distribution A. Infinite number of Data Gaussian Error Curve Concentration -40 -20 0 20 40 60 80 Occurrence 15 20 25 30 35 40 45 u = population mean σ = population std. dev
- 24. Gaussian Error Curve 68.3% of all data within ±1s 95.4% of all data within ± 2 s 99.7% of all data within ± 3s
- 25. Sample Mean Sample std dev (s) sample variance (s2 ) relative std deviation (rsd) N: # measurements Use calculator s / Mean * 100 (or 1000) Define: Reality: Finite data set Sum pts divide by Nx When N → ∞ , → u and s → σx
- 26. Important points: 1. In most situations, you will have a finite sample of data, therefore use standard deviation (s), not σ average students t 2. Use your calculator to calculate mean and std dev. Quicker and less error Useful advice: Know how to use your calculator x
- 27. Statistical Treatment of Random Errors B. Finite sample size Use statistics to set limits around a mean since can not determine the population mean Confidence intervals (4-2) 1. σ is not known (have a finite data set) t = Student’s t (Table 4-2) Memorize Degrees of Freedom: N-1 n ts x ±=µ
- 28. What does a confidence interval mean?
- 30. Statistical Treatment of Random Errors • Rejection of Data • Use caution Q-test • Simple and widely used Qexp = d / w = gap /range = l Xquest – Xnearest l / range Qexp > Qtable reject Memorize
- 32. If N = 4, mean = 5.0, and s = 1.0, what is confidence Interval (95%) Can you statistically reject 56.23 ? 55.95 56.00 56.04 56.08 56.23 Questions:
- 33. Method of Least Squares Calibration Curves y = mx + b Equation for a straight line memorize Useful advice: Examine your data for sensibility
- 34. Selectivity Sensitivity Specificity Linearity Blank Detection limit Standard addition Matrix effect Internal standard Important Terms