From: Subject: Error Propogation Date: Thu, 30 Oct 2003 20:51:18 -0500 MIME-Version: 1.0 Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Content-Location: http://mulliken.chem.hope.edu/~polik/Chem345-2000/errorpropagation.htm X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1165 Error = Propogation

ERROR=20 PROPAGATION

1.  Measurement of Physical=20 Properties

The = value of a=20 physical property often depends on one or more measured=20 quantities

Example:  Volume of a cylinder

 

2.  Systematic=20 Errors

A = systematic=20 error in the measurement of x, y, or z leads to an error in the = determination=20 of u.

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.

 

3.  Random=20 Errors

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.

If = the measured=20 variables are independent (non-correlated), then the cross-terms average = to=20 zero

as = dx, dy, and dz each take on both positive = and negative=20 values.

Thus,

Equating=20 standard deviation with differential, i.e.,

results in the=20 famous error propagation formula

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.

 

Thus, the=20 expected uncertainty in V is =B139=20 cm3.

4.  Purpose of Error=20 Propagation

=B7       =20 Quantifies precision of = results

 

Example:  V =3D 1131 =B1 39=20 cm3

=B7       =20 Identifies principle source of = error and=20 suggests improvement

 

Example:  Determine r better (not h!)

=B7       =20 Justifies observed standard = deviation

 

If sobserved=20 =BB scalculated = then the=20 observed standard deviation is accounted for

If sobserved = differs=20 significantly from scalculated = then perhaps=20 unrealistic values were chosen for sx, sy, and sz.

=B7       =20 Identifies type of error

 

If=20 =BDuobserrved = - = uliterature=BD =A3 scalculated = then error is=20 random error

If=20 =BDuobserrved = - = uliterature=BD >> scalculated = then error is=20 systematic error

5.  Calculating and Reporting = Values when=20 using Error Propagation

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 cm3

6.  Comparison of Error = Propagation to=20 Significant Figures

Use = of=20 significant figures in calculations is a rough estimate of error=20 propagation.

Example:

Keeping two=20 significant figures in this example implies a result of V =3D 1100 = =B1 100=20 cm3, which is much less precise than the result of V =3D 1131 =B1 39 cm3 = derived by error=20 propagation.

7.  Common Applications of the = Error=20 Propagation Formula

Several=20 applications of the error propagation formula are regularly used in = Analytical=20 Chemistry.

Example:

 

Example:

 

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.)