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Chapter 4 Slides

Chapter 4 Slides - Gaussian Distribution Chapter 4...

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1 Chapter 4 Statistics Dr. Elizabeth Binamira-Soriaga Department of Chemistry Texas A&M University Gaussian Distribution x = x i i = 1 n " n mean, x: Standard deviation, s: s = x i " x ( ) 2 i = 1 n # n " 1 For an infinite set of data, mean x approaches μ and s, ! (no systematic error). Recall variance and rsd Distribution ~ Gaussian for finite data with errors purely random Gaussian Curves μ = ± 1 " μ = ± 2 " μ = ± 3 " Range Percentage of Measurements 68.3 95.5 99.7 What fraction of observations in a Gaussian distribution is expected to be below μ - 3 ! ? Confidence Intervals Çonfidence interval expresses the certainty that the true population mean, μ , lies within a certain distance from the measured mean, x. The confidence interval of μ is: μ = x ± ts n s =measured standard deviation n = number of observations t = Student’s t Student’s t is a statistical tool to express confidence intervals and to compare results from different experiments to see if they are essentially the same Confidence interval: Values of Student’s t Degrees of freedom = n-1 Calculation of Confidence Intervals Find the 50% and 90% confidence intervals for carbohydrate content if the carbohydrate content of a glycoprotein is determined to be 12.6, 11.9, 13.0, 12.7 and 12.5 per 100 g of protein in replicate analyses First, calculate mean and standard deviation for set of data: x = 12.5 and s = 0.4 From Table 4-2, values of Student’s t are : 0.741 (50% confidence level and 2.132 (90% confidence level) Use t’s to solve for μ for each confidence level: μ = x ± ts n = 12.54 ± (0.741)(0.40) 5 = 12.5 ± 0.1 μ = x ± ts n = 12.54 ± (2.132)(0.40) 5 = 12.5 ± 0.3

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2 Calculation of Confidence Intervals Confidence Intervals (CI) as Estimates of Experimental Uncertainty x = 6.374 6 mL s = 0.001 8 mL (90% CI at +/- 0.001 7 mL) (90% CI at +/- 0.0007 mL) The Meaning of Confidence Intervals 50% CI: 50% of error bars would include μ 90% CI: 90% of error bars would include μ μ = 10000 and ! = 1000 Filled squares are data points that do not include the true population mean of 10000 Comparison of Means with Student’s t
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