The = value of a=20 physical property often depends on one or more measured=20 quantities
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Example: Volume of a cylinder
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A = systematic=20 error in the measurement of x, y, or z leads to an error in the = determination=20 of u.
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This is simply=20 the multi-dimensional definition of slope. =20 It describes how changes in u depend on changes in x, y, and z.
Example: A miscalibrated ruler results = in a=20 systematic error in length measurements. =20 The values of r and h must be changed by +0.1 = cm.
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Random errors=20 in the measurement of x, y, or z also lead to error in the = determination=20 of u. =20 However, since random errors can be both positive and negative, = one=20 should examine (du)^{2} rather than du.
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If = the measured=20 variables are independent (non-correlated), then the cross-terms average = to=20 zero
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as = dx, dy, and dz each take on both positive = and negative=20 values.
Thus,
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Equating=20 standard deviation with differential, i.e.,
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results in the=20 famous error propagation formula
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This expression=20 will be used in the Uncertainty Analysis section of every Physical = Chemistry=20 laboratory report!
Example: There is 0.1 cm uncertainty in = the ruler=20 used to measure r and h.
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Thus, the=20 expected uncertainty in V is =B139=20 cm^{3}.
=B7 =20 Quantifies precision of = results
Example: V =3D 1131 =B1 39=20 cm^{3}
=B7 =20 Identifies principle source of = error and=20 suggests improvement
Example: Determine r better (not h!)
=B7 =20 Justifies observed standard = deviation
If s_{observed}=20 =BB s_{calculated} = then the=20 observed standard deviation is accounted for
If s_{observed} = differs=20 significantly from s_{calculated} = then perhaps=20 unrealistic values were chosen for s_{x}, s_{y}, and s_{z}.
=B7 =20 Identifies type of error
If=20 =BDu_{obserrved} = - = u_{literature}=BD =A3 s_{calculated} = then error is=20 random error
If=20 =BDu_{obserrved} = - = u_{literature}=BD >> s_{calculated} = then error is=20 systematic error
Use = full=20 precision (keep extra significant figures and do not round) until the = end of a=20 calculation. Then keep = two=20 significant figures for the uncertainty and match precision for the=20 value.
Example: V =3D 1131 =B1 39 cm^{3}
Use = of=20 significant figures in calculations is a rough estimate of error=20 propagation.
Example:
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Keeping two=20 significant figures in this example implies a result of V =3D 1100 = =B1 100=20 cm^{3}, which is much less precise than the result of V =3D 1131 =B1 39 cm^{3} = derived by error=20 propagation.
Several=20 applications of the error propagation formula are regularly used in = Analytical=20 Chemistry.
Example:
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Example:
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Analytical=20 chemists tend to remember these common error propagation results, as = they=20 encounter them frequently during repetitive measurements. Physical chemists tend to = remember the=20 one general formula that can be applied to any case, as they encounter = widely=20 varying applications of error propagation. =20 (Or perhaps analytical chemists take a more utilitarian approach, = whereas=20 physical chemists take a more "from first principles"=20 approach.)