Given a formula with an absolute error in of , the absolute error is . The relative error is . If , then
(1)
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where denotes the mean, so the sample variance is given by
(2)
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(3)
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The definitions of variance and covariance then give
(4)
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(5)
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(6)
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(where ), so
(7)
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If and are uncorrelated, then so
(8)
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Now consider addition of quantities with errors. For , and , so
(9)
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For division of quantities with , and , so
(10)
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Dividing through by and rearranging then gives
(11)
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For exponentiation of quantities with
(12)
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and
(13)
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so
(14)
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(15)
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If , then
(16)
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For logarithms of quantities with , , so
(17)
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(18)
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For multiplication with , and , so
(19)
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(20)
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(21)
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For powers, with , , so
(22)
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(23)
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