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Chem. 31 – 2/6 Lecture Announcements I • Returning graded material in lab (diagnostic quizzes, quiz 1, lab procedures quiz) • Q1 scores were not great – (ave = 6.2 vs. 9.3 for Summer, ’16 which had part a) – It is your responsibility to learn material covered in class and in text homework problems (H2O2 example and 1.34/1.37) • SacCT has been updated (has quiz scores and quiz solutions) • Pipet Calibration Lab Report due next Monday Announcements II • Additional Problem 1 – due 2/15 • Today’s Lecture – Error and Uncertainty (Chapter 3) • Propagation of Uncertainty – Mixed operation problems – Gaussian Statistics and Calibration (Chapter 4) • • • • • Mean value and standard deviation Gaussian distributions for populations Application to measurement statistics Z-value problems - % between limits Confidence Intervals (if time) Propagation of Uncertainty • Calculation of the density of a liquid: Density = mliquid/Vliquid with mass measured as weight gain in a container (mfull – mempt) Chapter 4 Calculation of Average and Standard Deviation • Average x x i n • Standard Deviation x x 2 Sx i n 1 Note: You are welcome to use function keys on your calculator to calculate average and standard deviation Chapter 4 – Gaussian Distributions • • Gaussian Distributions are often observed when sample size gets large Example 1: repeated measurement of MMP conc. μ Sample vs. Population: Sample mean x Sample standard deviation S population mean μ population standard deviation σ x MMP Conc. 6.00 5.80 5.60 MMP Conc. (ppm) • MMP = methylmannopyranoside = interal standard added at a conc. of 5.00 ppm 5.40 5.20 S 5.00 4.80 4.60 4.40 4.20 4.00 1 3 5 7 9 11 13 15 17 Sample Number • As sample size increases: x →μ Also note that mean values vary less than individual measurements (see 4 measurement means) and S → σ 19 21 23 25 Chapter 4 – Gaussian Distributions MMP Concentration Distribution 8 7 Number of Values 6 5 Actual Expected from Data 4 3 2 1 0 4.0-4.25 4.25-4.5 4.5-4.75 4.75-5.0 5.0-5.25 5.25-5.5 5.5-5.75 Conc. Range 5.75-6 Chapter 4 – Gaussian Distributions • Second Example: Mass Spectrometer Measurements (x-axis is mass to charge ratio or mass for +1 ions) (1) Spec #1 * [BP = 234.1, 52707] 100 1343.9877 90 80 70 % Intensity 60 50 40 1344.9770 30 20 10 0 1 3 4 3 .1 2 3 6 0 1 3 4 3 .9 2 6 3 6 1 3 4 4 .7 2 9 1 3 1 3 4 5 .5 3 1 8 9 m/z 1 3 4 6 .3 3 4 6 6 Chapter 4 – Gaussian Distributions Idealized distribution occurs as n →∞ • Math for Gaussian Distribution: 1 e ( x ) 2 0.45 0.4 / 2 Normalized Gaussian Distribution: μ = 0 and σ = 1 Use of Normalized Gaussian Distribution: 0.35 0.3 Frequency y 2 Normal Distribution 0.25 0.2 0.15 0.1 0.05 0 -5 -4 -3 -2 -1 0 1 2 X or Z value Area under curve gives probability of finding value between limits 3 4 5 Chapter 4 – Gaussian Distributions Examples: Normal Distribution -∞<Z<∞ Area = 1 0.45 0.4 Since curve is symmetrical, 0 < Z < 1.5 Area = 0.433 (See Table 4-1 in text) 0.3 Frequency 0 < Z < ∞ Area = 0.5 0.35 0.25 0.2 0.15 0.1 0.05 0 -5 -4 -3 -2 -1 0 1 2 3 4 X or Z value Note: non-normal distributions can be converted to normal distributions as follows: Z = (x - μ)/σ 5 Chapter 4 – Gaussian Distributions Now for limit problems – example 1 – population statistics: A lake is stocked with trout. A biologist is able to randomly sample 42 fish in the lake (and we can assume that 42 fish are enough for proper – Z-based statistics). Each fish is weighed and the average and standard deviation of the weight are 2.7 kg and 1.1 kg, respectively. If a fisherman knows that the minimum weight for keeping the fish is 2.0 kg, what percent of the time will he have to throw fish back? (assuming catching is not size-dependent) 1st part: convert limit (2.0 kg) to normalized (Z) value: Z = (x – )/ 2nd part: use Z area to get percent Chapter 4 – Gaussian Distributions Limit problem – example 2 – measurement statistics: A man wants to get life insurance. If his measured cholesterol level is over 240 mg/dL (2,400 mg/L), his premium will be 25% higher. His level is measured and found to be 249 mg/dL. His uncle, a biochemist who developed the test, tells him that a typical standard deviation on the measurement is 25 mg/dL. What is the chance that a second measurement (with no crash diet or extra exercise) will result in a value under 240 mg/dL (e.g. beat the test)? Graphical view of examples Equivalent Area Frequency Normal Distribution 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Table area Desired area -5 -4 -3 -2 -1 0 1 2 3 4 5 Z value 240 249 X-axis Chapter 4 – Calculation of Confidence Interval 1. 2. x n Z depends on area or desired probability At Area = 0.45 (90% both sides), Z = 1.65 At Area = 0.475 (95% both sides), Z = 1.96 => larger confidence interval Normal Distribution Frequency Confidence Interval = x + uncertainty Calculation of uncertainty depends on whether σ is “well known” 3. When is not well known (covered later) 4. When is well known (not in text) Value + uncertainty = Z 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -3 -2 -1 0 Z value 1 2 3 Chapter 4 – Calculation of Uncertainty Example: The concentration of NO3- in a sample is measured 2 times and found to give 18.6 and 19.0 ppm. The method is known to have a constant relative standard deviation of 2.0% (from past work). Determine the concentration and 95% confidence interval. Chapter 4 – Calculation of Confidence Interval with Not Known Value + uncertainty = tS x n t = Student’s t value t depends on: - the number of samples (more samples => smaller t) - the probability of including the true value (larger probability => larger t) Chapter 4 – Calculation of Uncertainties Example • Measurement of lead in drinking water sample: – values = 12.3, 9.8, 11.4, and 13.0 ppb • What is the 95% confidence interval?