Lecture_Statistics_Spring_2013b

Ii means and standard deviations properties of the

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Unformatted text preview: ound a mean value. II. Means and Standard Deviations Questions: (1)  Why is (Xi- x ) squared? (2)  Why is it N-1 in the denominator for samples and N for populations? II. Means and Standard Deviations Properties of the mean and the variance *Mean µ aX + b = aµ X + b µ aX = aµ X µX+b = µX + b *Variance X: Variable a and b: Constants 2 σaX = a 2 σ 2 X σ2 + b = σ2 X X 2 σ aX + b = a 2 σ 2 X 2 σ aX + bY = a 2 σ 2 + b 2 σ 2 X Y X, Y: Variable a and b: Constants II. Means and Standard Deviations Standard Deviation For populations ȹ N 2 ȹ σ = ȹ ∑ (x i − µ ) ȹ N ȹ ȹ ȹ i =1 Ⱥ For samples ȹ N 2 ȹ s = ȹ ∑ (x i − x ) ȹ ȹ ȹ ȹ i =1 Ⱥ (N − 1) * The standard deviation has the same units as the mean. * Standard deviations are not additive for independent variables. * The standard deviation has the following properties: σ aX = a σ X σ aX + b = a σ X σX + b = σX σ aX + bY = a 2 σ 2 + b 2 σ 2 X Y III. Probability Distributions & Confidence Intervals Gaussian Distribution p ( x ; µ, σ ) = Ⱥ (x − µ )2 Ⱥ expȺ− Ⱥ 2 2 πσ 2σ Ⱥ Ⱥ Ⱥ Ⱥ 1 * Key feature of a Gaussian probability distribution function: Ⱥ (x − µ )2 Ⱥ expȺ− Ⱥ dx = 1 ∫ 2 2 πσ 2σ Ⱥ Ⱥ −∞ Ⱥ Ⱥ ∞ 1 * Similarly, the integral from negative infinity to X = A is the probability of observing a value less than or equal to A: Ⱥ (x − µ )2...
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