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### Chapter 4-3811-stats

Course: CHEM 3811, Fall 2009
School: Clayton
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Word Count: 855

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CHEMISTRY ANALYTICAL CHEM 3811 CHAPTERS 4 DR. AUGUSTINE OFORI AGYEMAN Assistant professor of chemistry Department of natural sciences Clayton state university CHAPTER 4 STATISTICS STATISTICS - Random errors are always associated with measurements - No conclusion can be drawn with complete certainty - Scientists use statistics to accept conclusions that have high probability of being correct and to reject...

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CHEMISTRY ANALYTICAL CHEM 3811 CHAPTERS 4 DR. AUGUSTINE OFORI AGYEMAN Assistant professor of chemistry Department of natural sciences Clayton state university CHAPTER 4 STATISTICS STATISTICS - Random errors are always associated with measurements - No conclusion can be drawn with complete certainty - Scientists use statistics to accept conclusions that have high probability of being correct and to reject conclusions that have low probability of being correct - Statistics deals with only random errors - Systematic errors should be detected and eliminated THE GAUSSIAN DISTRIBUTION - Symmetric bell shaped curve representing the distribution of experimenal data - Characterized by mean and standard deviation - The Gaussian function is f(x) = ae (x ) 2 2 2 1 - a is the height of the curves peak, a = 2 - is the position of the center of the peak (the mean) - is a measure of the width of the curve (standard deviation) THE GAUSSIAN DISTRIBUTION THE GAUSSIAN DISTRIBUTION Arithmetic Mean - Also known as the average - Is the sum of the measured values divided by the number of measurements _ x = x i i n 1 = ( x 1 + x 2 + x 3 +..... + x n ) n = sigma, symbol for the sum xi = a measured value n = number of measurements THE GAUSSIAN DISTRIBUTION Standard Deviation - A measure of the width of the distribution - Small standard deviation gives narrow distribution curve s = x i x i n 1 _ 2 xi = a measured value n = number of measurements n-1 = the degrees of freedom Standard deviation has the same number of decimal places as the mean THE GAUSSIAN DISTRIBUTION Relative Standard Deviation Relative Standard Deviation = s x _ x 100 - The sample mean and sample standard deviation (x and s) apply to finite set of measurements - For infinite set of data use is made of the true mean or population mean (designated ) and the true standard deviation (designated ) THE GAUSSIAN DISTRIBUTION Variance - Is the square of the standard deviation - Variance = 2 THE GAUSSIAN DISTRIBUTION Median - The middle number in a series of measurements arranged in increasing order - The average of the two middle numbers if the number of measurements is even THE GAUSSIAN DISTRIBUTION Mode - The value that occurs the most frequently Range - The difference between the highest and the lowest values All but the range should be expressed with one extra digit beyond the last significant digit THE GAUSSIAN DISTRIBUTION Probability - Range of measurements for ideal Gaussian distribution - The percentage of measurements lying within the range given (one, two, or three standard deviation on either side of the mean) Range 1 2 3 Gaussian Distribution (%) 68.3 95.5 99.7 CONFIDENCE INTERVAL - Range of values within which there is a specified probability of finding the true mean () - The t-test is used to express confidence intervals - Also used to compare results from different experiments CONFIDENCE INTERVAL ts = x n _ - That is, the range of confidence interval is ts/n below the mean and + ts/n above the mean - For better precision reduce confidence interval by increasing number of measurements Refer to table 4-2 on page 87 for t-test values CONFIDENCE INTERVAL To test for comparison of Means - Calculate the pooled standard deviation (spooled) - Calculate t - Compare the calculated t to the value of t from the table - The two results are significantly different if the calculated t is greater than the tabulated t at 95% confidence level CONFIDENCE INTERVAL For two sets of data with - n1 and n2 measurements - averages of x1 and x2 - standard deviations of s1 and s2 s pooled = 2 s1 ( n1 1) + s 2 ( n 2 1) 2 n1 + n 2 2 t = x1 x 2 s pooled n1n 2 n 1 +n 2 Degrees of freedom = n1 + n2 - 2 GRUBBS TEST FOR AN OUTLIER - An outlier is a datum that is far from other points - Grubbs test is used to determine whether an outlier should be rejected or retained G= questionable value x s GRUBBS TEST FOR AN OUTLIER - Calculate mean, standard deviation, and then G - Compare G calculated to G tabulated (Table 4-4 on page 92) - The questionabl...

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Clayton - CHEM - 3811
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