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| Propagation of Errors |
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| Final result of an experiment is calculated from a number of observations taken from different instruments, connected through a formula. |
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| Result involving sum of two observed quantities |
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| X is the sum of 2 observed quantities a and b. |
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| X = a + b |
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| Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b |
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| Result involving difference of two observed quantities |
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| Suppose X = a - b |
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| Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be maximum error in X. |
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 Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b |
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| From equations (1) and (2) it is evident that, when result involves sum or difference of 2 observed quantities, absolute error is the sum of absolute errors in the observed quantities. |
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| Result involving the product of two observed quantities |
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| Suppose X = ab |
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| Let Da and Db be absolute errors in measurements of quantities a and b, values of a and b and DX be the maximum possible error in X. |
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| Dividing both sides by X = ab, we get |
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 are
relative errors of fractional errors in values of a, b and x. Neglecting
as its product is very small.
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| The above result is obtained by logarithmic differentiation. |
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| Take log on both sides, |
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| Log X = log a + log b |
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| Differentiating, we get , |
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| Thus, maximum relative error in X = maximum relative error in a x maximum relative error in b |
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| Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b |
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| Result involving quotient of 2 observed quantities |
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| Let Da and Db be absolute errors in measurement of quantities a and b and DX be maximum possible error in X. |
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| Maximum possible relative error in X, |
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| Maximum relative error in X = maximum relative error in a + maximum relative error in b |
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| Maximum percentage error in X, |
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| i.e., Maximum percentage error in X = maximum percentage error in a + maximum percentage in b. From equations 3, 4, 5 and 6, it is seen that when the result involves the multiplication or quotient of 2 observed quantities, the maximum possible relative error in the result is equal to the sum of the relative errors in the observed quantities. |
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| Result involving product of powers of observed quantities |
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| Relative error in an is n times the relative error a |
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| It can be proved that maximum relative error in X, |
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| Also, maximum percentage error in X, |
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| Maximum percentage error in X = l times maximum percentage error in a + m times maximum percentage error in b + n times maximum percentage error in c. |
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