Error Analysis

Stub article, work in progress

A core reference work for error analysis is Data Reduction and Error Analysis for the Physical Sciences, by Philip Bevington and D. Keith Robinson.

A few core results are presented below for easy reference.

The equations assume the combination of independent, normally distributed, uncorrelated errors arising from various terms $ x_i $ combined in different ways. Frequently the distribution of an error term is not normal, e.g. the Poisson distribution, or is unknown. But the impact of even a very different distribution, for instance the Uniform distribution, is often very small on the combination of errors according to the formal results for the Normal distribution.

Addition or Subtraction

The error of the sum of quantities $ x_i $ is equal to the square root of the sum of the squared standard errors. We say that errors add in quadrature in this definition. For

\[ x_{\rm tot} = \sum_i x_i \]

The standard error $\sigma_{x_{\rm tot}} $ is defined by:

\[ \sigma_{x_{\rm tot}} = \sqrt{ \sum_i (\sigma_{x_i})^2    } \]

It does not matter whether the data is added or subtracted -- uncorrelated errors always add in quadrature.

Multiplication or Division

The error of a product or ratio of some collection of independent variables $ x_i $ with uncorrelated errors grows by adding the proportional errors in quadrature. That is, for:

\[   x_{\rm prod} = \prod_i x_i   \]

The proportional error in the total will be:

\[   \frac{\sigma_{x_{\rm prod}}}{x_{\rm prod}} = \sqrt{  \sum_i \left( \frac{\sigma_{x_i}}{x_i} \right)^2     }  \]

Once again, this applies whether or not the terms are multiplied or divided. Note that once the proportional error exceeds unity, little value can be placed in the resultant.

Products of powers

The error of a product or ratio of some collection of independent variables $ x_i^{m_i} $ with uncorrelated errors grows by adding the proportional errors in quadrature, with each term multiplied by the value of the exponent. That is, for:

\[   x_{\rm prod} = \prod_i x_i^{m_i}   \]

The proportional error in the total will be:

\[   \frac{\sigma_{x_{\rm prod}}}{x_{\rm prod}} = \sqrt{  \sum_i \left( \frac{m_i  \cdot \sigma_{x_i^{m_i}}}{x_i^{m_i}} \right)^2     }  \]

As before, this applies whether or not the terms are multiplied or divided. Note again that once the proportional error exceeds unity, little value can be placed in the resultant.


-- DonBarry - 2014-09-14

 
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