The value of a physical property often depends on one or more measured quantities
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Example: Volume of a cylinder

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A systematic error in the measurement of x, y, or z leads to an error in the determination of u.
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This is simply the multidimensional definition of slope. It describes how changes in u depend on changes in x, y, and z.
Example: A miscalibrated ruler results in a systematic error in length measurements. The values of r and h must be changed by +0.1 cm.

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Random errors in the measurement of x, y, or z also lead to error in the determination of u. However, since random errors can be both positive and negative, one should examine (du)^{2} rather than du.
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If the measured variables are independent (noncorrelated), then the crossterms average to zero
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as dx, dy, and dz each take on both positive and negative values.
Thus,
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Equating standard deviation with differential, i.e.,
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results in the famous error propagation formula
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This expression will be used in the Uncertainty Analysis section of every Physical Chemistry laboratory report!
Example: There is 0.1 cm uncertainty in the ruler used to measure r and h.

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Thus, the expected uncertainty in V is ±39 cm^{3}.
· Quantifies precision of results

Example: V = 1131 ± 39 cm^{3} 
· Identifies principle source of error and suggests improvement

Example: Determine r better (not h!) 
· Justifies observed standard deviation

If s_{observed} » s_{calculated} then the observed standard deviation is accounted for If s_{observed} differs significantly from s_{calculated} then perhaps unrealistic values were chosen for s_{x}, s_{y}, and s_{z}. 
· Identifies type of error

If ½u_{obserrved}  u_{literature}½ £ s_{calculated} then error is random error If ½u_{obserrved}  u_{literature}½ >> s_{calculated} then error is systematic error 
Use full precision (keep extra significant figures and do not round) until the end of a calculation. Then keep two significant figures for the uncertainty and match precision for the value.
Example: V = 1131 ± 39 cm^{3}
Use of significant figures in calculations is a rough estimate of error propagation.
Example: 
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Keeping two significant figures in this example implies a result of V = 1100 ± 100 cm^{3}, which is much less precise than the result of V = 1131 ± 39 cm^{3} derived by error propagation.
Several applications of the error propagation formula are regularly used in Analytical Chemistry.
Example: 
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Example: 
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Analytical chemists tend to remember these common error propagation results, as they encounter them frequently during repetitive measurements. Physical chemists tend to remember the one general formula that can be applied to any case, as they encounter widely varying applications of error propagation. (Or perhaps analytical chemists take a more utilitarian approach, whereas physical chemists take a more "from first principles" approach.)