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Date: 7-2-2021
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Date: 4-4-2021
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Date: 23-4-2021
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Given a formula with an absolute error in
of
, the absolute error is
. The relative error is
. If
, then
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(1) |
where denotes the mean, so the sample variance is given by
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(2) |
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(3) |
The definitions of variance and covariance then give
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(4) |
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(5) |
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(6) |
(where ), so
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(7) |
If and
are uncorrelated, then
so
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(8) |
Now consider addition of quantities with errors. For ,
and
, so
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(9) |
For division of quantities with ,
and
, so
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(10) |
Dividing through by and rearranging then gives
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(11) |
For exponentiation of quantities with
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(12) |
and
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(13) |
so
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(14) |
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(15) |
If , then
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(16) |
For logarithms of quantities with ,
, so
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(17) |
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(18) |
For multiplication with ,
and
, so
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(19) |
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(20) |
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(21) |
For powers, with ,
, so
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(22) |
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(23) |
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 14, 1972.
Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, pp. 58-64, 1969.
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