Lecture_Statistics_Spring_2013b

Propagation of error example 1 consider the perimeter

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Unformatted text preview: ȹ ȹ ȹ ∂X ȹ ȹ ȹ ∂X1 Ⱥ X 2 ,…, X N ȹ ∂X 2 Ⱥ X1 , X 3 ,…, X N ȹ N Ⱥ X1 , X N −1 2) Square of above Equation: Ⱥ Ⱥ ȹ ∂Q ȹ ȹ ∂Q ȹ 2 Ⱥȹ ∂Q ȹ (dQ ) = ȹ ȹ dX1 + ȹ dX 2 + … + ȹ dX N Ⱥ ȹ ∂X ȹ ȹ ∂X ȹ ȹ ȹ ∂X ȹ ȹ Ⱥȹ 1 Ⱥ X ,…, X Ⱥ ȹ 2 Ⱥ X1 , X 3 ,…, X N ȹ N Ⱥ X1 , X N−1 2 N Ⱥ Ⱥ 2 IV. Propagation of Error 2 σ Q = (dQ )2 = 2 Ⱥȹ ∂Q ȹ 2 ȹ ȹ ∂Q ȹ Ⱥȹ ȹ ȹ (dX 2 )2 + … + ȹ ∂Q (dX1 ) 2 + ȹ ȹ ∂X ȹ ȹ ∂X Ⱥȹ ∂X1 ȹ X ,…, X ȹ Ⱥ 2 ȹ 2 Ⱥ X1 , X 3 ,…, X N ȹ N N Ⱥ 2 Ⱥ ȹ 2 Ⱥ ȹ (dX N ) ȹ Ⱥ Ⱥ X1 , X N −1 Ⱥ Noting that 1) the dXi2 are variances, i.e. s2Xi, 2) the dX1, dX2, etc. are sX1, sX2, etc., respectively, 3) dX1dX2, dX1dX3, etc. are covariances, which are zero if the Xi are independent. 2 2 Ⱥȹ ∂Q ȹ 2 Ⱥ ȹ ∂Q ȹ ȹ ∂Q ȹ 2 2 2 2 Ⱥ σ Q = Ⱥȹ σ X + ȹ σ X + … + ȹ σX ȹ ∂X ȹ ȹ ȹ ∂X ȹ ȹ ȹ ∂X ȹ ȹ 1 2 N Ⱥ Ⱥȹ 1 Ⱥ X ,…, X ȹ 2 Ⱥ X1 , X 3 ,…, X N ȹ N Ⱥ X1 , X N−1 2 N Ⱥ Ⱥ Eq* σQ = N ȹ ∂Q ȹ ∑ ȹ ȹ i =1ȹ X i 2 ȹ σ 2 ȹ Xi Ⱥ X j *j represents all other N – 1 independent variables IV. Propagation of Error Example 1 Consider the perimeter of a right triangle which has...
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This note was uploaded on 01/26/2014 for the course CHEM 3625 taught by Professor Mrjohnson during the Spring '08 term at Virginia Tech.

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