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Error_Propagation

# Error_Propagation - APPENDI X 2 PROPAGATI ON OF ERROR If...

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Chemistry 223 – Appendix 2 1 APPENDIX 2: PROPAGATION OF ERROR If several measured values are combined in a calculation to yield a result, then it is necessary to combine the uncertainties in the individual measured values to obtain the uncertainty in the calculated result. This is called propagation of error (or uncertainty). Suppose you have a function, F , composed of several experimental variables x, y, z,... . If the errors x , y , and z are given by the standard deviations sx, sy, sz , ... and those errors are random and independent, then the variance in the function is given approximately by, s F 2 = " F " x # \$ % & ( 2 s x 2 + " F " y # \$ % & ( 2 s y 2 + " F " z # \$ % & ( 2 s z 2 + ... (1) where it is assumed that the partial derivatives with respect to each variable are known or can obtained. From eqn. (1) the error (uncertainty) associated with any given measured function can be calculated. However, the specific forms given below will suffice for the vast majority of cases encountered in practice. I.A. Addition and Subtraction Let F = x + y " z (2) then s F 2 = s x 2 + s y 2 + s z 2 (3) or s F = s x 2 + s y 2 + s z 2 ( ) 1 / 2 (4) where we make use of the fact that the standard deviation, s , is the square root of the variance, s 2 . So, as shown above for addition and subtraction, the overall absolute

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Error_Propagation - APPENDI X 2 PROPAGATI ON OF ERROR If...

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