7 - Module 3 Data Treatment Statistical Analysis Gaussian...

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1 Module 3 Data Treatment • Statistical Analysis – Gaussian distribution; sample and population; sample mean and population mean; sample standard deviation and the population standard deviation; relative standard deviation (RSD) and coefficient of variation (CV) – Confidence limits; Student’s “t” values; comparison of experimental mean with true value; pooling standard deviations; comparison of two experimental means; detection of gross errors and rejection tests. 1 New terms: Error analysis Sample Variance ( s 2 ): – Square of the sample deviation ( s ) –= ( X i –X) 2 ( N –1) 1) - (N ) X - X ( 2 i s 2 2 2 Application of statistics Significance of the standard deviation – Statistical equations are based on measurements of very large numbers of data – If a measurement is repeated thousands of times we can plot the DISTRIBUTION of results as: 3
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2 Standard deviation uency 4 -3 σ -2 σ -1 σ 0 +1 σ +2 σ +3 σ d i = x i μ Relative freq Standard deviation Significance of the standard deviation – Detailed results show that when we calculate the standard deviation, we can predict the following if the result is repeated many times: • 68% of results will be within 1 standard deviation ( 1 σ ) of the mean • 96% of results will be within 2 standard deviations ( 2 σ ) of the mean • 99.7% of results will be within 3 standard deviations ( 3 σ ) of the mean 5 Application of statistics Confidence intervals, confidence limits – Goal for sample statistics is to determine population values σ can be approximated by s More difficult to determine μ • More difficult to determine – We can define a range of values which will ‘‘probably’’ include μ • Probably in a statistical sense • Based on Gaussian (normal) distribution of data – Called the ‘confidence interval’ ( CI ) 6
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3 Application of statistics Confidence intervals, confidence limits – Knowing the standard deviation, we can predict Confidence limits for our set of results – Confidence limits give ranges in which we can quote our results with a stated degree of quote our results with a stated degree of confidence or confidence level (CL) (e.g. 95%) • Meaning that the result, if repeated, would fall within the given range on this number of times out of 100 7 Confidence limits 60.53, 60.53, 60.37, 60.37, 60.29, 60.29 % Cl
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7 - Module 3 Data Treatment Statistical Analysis Gaussian...

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