Lecture_Statistics_Spring_2013b

Lecture_Statistics_Spring_2013b - Lecture One Statistical...

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Statistical Treatment of Data (Error Analysis in the Physical Chemistry Laboratory) Bingbing Li and Alan R. Esker CHEM 3625 Virginia Tech Lecture One
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Outline Accuracy and Precision Means and Standard Deviations Probability Distributions and Confidence Intervals Propagation of Error Significant Figures Linear Regression Reference
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I. Accuracy and Precision Statistics definition: The true value is the mean (or average ) of the sample population, which is composed of all possible members of the group . Hence, estimate of the mean obtained from a finite subset of population may be in error. Two categories of the main sources of such errors: * Systematic error : arises from bias that is placed on the measurement either by the instrument itself or by an improper method of reading or using the instrument. * Random error: arises from the intrinsic limitation in the sensitivity of the instrument or measuring device and in our ability to interpret the instrument's output. Random errors may represent fundamental limitations on our ability to perceive reality. Other types of error: “blunder” and “model error”
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± Systematic error vs. Accuracy Systematic errors prevent one from obtaining the true value. * Random error vs. Precision The measurement has high precision if the random errors are small . There is no necessary relationship between accuracy and precision. Precision DOES NOT necessarily imply accuracy, since an experiment with small random errors may still give inaccurate results due to large systematic errors. See Example in the next slide I. Accuracy and Precision
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Three targets for three different guns and three different marksmen. Target A : reasonable accuracy and precision. Target B : poor precision, although the average is about the center of the target. Target C : excellent precision, but poor accuracy. A B C × × × × × × × × × I. Accuracy and Precision
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II. Means and Standard Deviations A population represents every member of a group Population Mean, μ Population Variance, σ 2 A sample represents some subset of a population N x N 1 i i Ο Ο Π Ξ Μ Μ Ν Λ = μ = ( ) N x N 1 i 2 i 2 Ο Ο Π Ξ Μ Μ Ν Λ μ = σ = 2 2 2 x x = σ Sample Mean, x Sample Variance, s 2 N x x N 1 i i Ο Ο Π Ξ Μ Μ Ν Λ = = ( ) ( ) 1 N x x s N 1 i 2 i 2 Ο Ο Π Ξ Μ Μ Ν Λ = = Ο Π Ξ Μ Ν Λ Ο Π Ξ Μ Ν Λ = σ = 2 2 2 2 x x 1 N N 1 N N s Variance describes the distribution of values around a mean value.
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Questions: (1) Why is (X i - ) squared? (2) Why is it N-1 in the denominator for samples and N for populations? x II. Means and Standard Deviations
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Properties of the mean and the variance ± Mean Variance X aX a μ = μ b X b X + μ = μ + b a X b + μ = μ + X: Variable a and b: Constants 2 X 2 2 a σ = σ 2 X 2 b X σ = σ + 2 X 2 2 b a σ = σ + 2 Y 2 2 X 2 2 bY b a σ + σ = σ + X, Y: Variable a and b: Constants II. Means and Standard Deviations
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Standard Deviation For populations For samples ( ) N x N 1 i 2 i Ο Ο Π Ξ Μ Μ Ν Λ μ = σ = ( ) ( ) 1 N x x s N 1 i 2 i Ο Ο Π Ξ Μ Μ Ν Λ
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Lecture_Statistics_Spring_2013b - Lecture One Statistical...

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