Lecture 4,5. Statistics Chpt4M

Lecture 4,5. Statistics Chpt4M - Gaussian or Normal...

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Unformatted text preview: Gaussian or Normal Distribution Gaussian Mean å x= i xi n = 1 ( x1 + x 2 + x 3 + ... + x n ) n Standard Deviation å s= i ( x i - x) n-1 2 Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Gaussian or Normal Distribution Gaussian Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. ed., Daniel C. Harris, Exploring Chemical Daniel Analysis, 3rd ed., W.H. Freeman and Analysis, Company, New York, 2005. Company, Fig. 4.2 Can the standard deviation be improved by more measurements? more Confidence Intervals Confidence µ= x± ts n • From a limited number of measurements, it is impossible to find the true mean and true standard deviation. deviation. • The confidence interval is a range of values within which there is a specified probability of finding the true mean. true Confidence Intervals Confidence µ= x± ts n Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Confidence Intervals Confidence In replicate analyses, the carbohydrate In content of a glycoprotein is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein. Find the 50% and 90% confidence intervals for the carbohydrate content. the Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Can the confidence interval be made more narrow by more measurements? narrow Confidence Intervals Confidence Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Comparing Means Comparing t= x1 - x 2 spooled n1n2 n1 + n2 spooled 2 s1 (n1 - 1)+ s2 (n2 - 1) 2 = n1 + n2 - 2 degrees of freedom = n1 + n2 - 2 If tcalc > ttable, then the difference is significant Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Comparing Means Comparing t= x1 - x 2 spooled n1n2 n1 + n2 spooled 2 s1 (n1 - 1)+ s2 (n2 - 1) 2 = n1 + n2 - 2 If tcalc > ttable, then the difference is significant Here: t = 20.2; degrees of freedom = ? Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Confidence Intervals Confidence µ= x± ts n 12 10 8 6 4 2 0 t for 95% confidence interval 0 5 10 15 20 25 Grubbs Test Grubbs For Outliers G= questionable value - x s Daniel C. Harris, Exploring Chemical Analysis, 4th ed., W.H. Daniel Freeman and Company, New York, 2005. Freeman The Q Test The gap Q= range 12 0.38 Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. Sons, Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Daniel Freeman and Company, New York, 2005. Freeman Do You Really Have a Gaussian Distribution? Do Your data does not always follow a Gaussian Your distribution. distribution. 0.4 0.20 0.3 0.15 0.2 0.10 0.1 0.05 2 4 6 8 2 4 6 8 10 Method of Least Squares Method Residual: di = yi – y = yi – (mxi + b) di2 = (yi – y)2 = (yi – mxi - b)2 The “best” fit line minimizes the sum of the squares. Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 2004. Method of Least Squares Method The “best” fit line minimizes the sum of the squares. Assumptions: 1) All data points have the same error. 2) The error is random and Gaussian. Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 2004. Method of Least Squares Method 6 5 4 3 2 1 1 2 3 4 5 The “best” fit line minimizes the sum of the squares. Assumptions: 1) All data points have the same error. 2) The error is random and Gaussian. Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 2004. Method of Method Least Squares: Linear Linear Regression Regression m= nå (x i y i ) nå x 2 i å xå (å x ) i i yi 2 β = y - mx Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 2004. Calibration Curves Calibration Can we really Can discard that questionable point? questionable Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Grubbs Test Grubbs For Outliers G= questionable value - x s Daniel C. Harris, Exploring Chemical Analysis, 4rd ed., W.H. Daniel Freeman and Company, New York, 2005. Freeman Calibration Curves Calibration Can we really Can discard that questionable point? questionable Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman Method of Method Least Squares: Linear Linear Regression Regression m= nå (x i y i ) nå x 2 i å xå (å x ) i i yi 2 β = y - mx Gary D. Christian, Analytical Chemistry, 6th ed., John Wiley & Sons, Inc., U.S.A., 2004. 2004. Uncertainty in Least Squares Parameters Uncertainty sy = å (d i ) n-2 n 2 sm = sy n å x - (å x i ) 2 i 2 sb = sy å 2 i x2 i 2 nå x - (å x i ) Coefficient of Determination (r2) Coefficient r2 = 1 - å å (d i ) 2 2 ( y i - y) r describes the fraction of the variation in y that can describes be explained by a linear relationship with x. be r is called the correlation coefficient. Coefficient of Determination (r2) Coefficient QuickTimeª and a decompressor are needed to see this picture. Distributions and their respective r values {from wikipedia.org} Distributions => Do not overinterpret r2! => r2 takes values between 0 and 1. A high correlation will be characterized by large r2 value. value. Sometimes a small r2 can be misleading. Calibration Curves Calibration m = 0.01630 µ g-1 b = 0.0047 sy = 0.0059 sm = 0.00022 µ g-1 sb = 0.0026 Daniel C. Harris, Exploring Chemical Analysis, 3rd ed., W.H. Freeman and Company, New York, 2005. Freeman y value of unknown = 0.373 Finding the Unknown: Finding y = mx + b y-b =x m solve for x => sx = sy m (y - y) 11 ++ k n m2 å ( x - x ) 2 i 2 not ( y i - y ) 2 m: slope k: number of measurements of unknown n: number of measurements of calibration samples ...
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