ERROR PROPAGATION

1. Measurement of Physical Properties

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

Example: Volume of a cylinder

2. Systematic Errors

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

This is simply the multi-dimensional 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.

3. Random Errors

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.

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

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

Thus,

Equating standard deviation with differential, i.e.,

results in the famous error propagation formula

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.

Thus, the expected uncertainty in V is ±39 cm³.

4. Purpose of Error Propagation

· Quantifies precision of results

Example: V = 1131 ± 39 cm³

· 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_literature½ £ s_calculated then error is random error

If ½u_obserrved - u_literature½ >> s_calculated then error is systematic error

5. Calculating and Reporting Values when using Error Propagation

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 cm³

6. Comparison of Error Propagation to Significant Figures

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

Example:

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

7. Common Applications of the Error Propagation Formula

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

Example:

Example:

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.)