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==Applications==

===Global positioning system===
{{Main|Global positioning system}}
The '''analysis of errors computed using the [[global positioning system]]''' is important for understanding how GPS works, and for knowing what magnitude errors should be expected. The Global Positioning System makes corrections for receiver clock errors and other effects but there are still residual errors which are not corrected. The Global Positioning System (GPS) was created by the United States Department of Defense (DOD) in the 1970s. It has come to be widely used for navigation both by the U.S. military and the general public.

===Molecular dynamics simulation ===

In [[molecular dynamics]] (MD) simulations, there are errors due to inadequate sampling of the phase space or infrequently occurring events, these lead to the statistical error due to random fluctuation in the measurements.

For a series of {{mvar|M}} measurements of a fluctuating property {{mvar|A}}, the mean value is:

<math display=block> \langle A \rangle = \frac{1}{M} \sum_{\mu=1}^M A_{\mu}. </math>

When these {{mvar|M}} measurements are independent, the variance of the mean {{math|1={{angbr|''A''}} }} is:

<math display=block> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}( A  ), </math>

but in most MD simulations, there is correlation between quantity {{mvar|A}} at different time, so the variance of the mean {{math|1={{angbr|''A''}} }} will be underestimated as the effective number of independent measurements is actually less than {{mvar|M}}. In such situations we rewrite the variance as:

<math display=block> \sigma^{2}( \langle A \rangle ) = \frac{1}{M} \sigma^{2}(A) \left[ 1 + 2 \sum_\mu \left( 1 - \frac{\mu}{M} \right) \phi_{\mu} \right],</math>

where <math>\phi_{\mu}</math> is the [[autocorrelation function]] defined by

<math display=block> \phi_{\mu} = \frac{ \langle A_{\mu}A_{0} \rangle - \langle A \rangle^{2} }{ \langle A^{2} \rangle - \langle A \rangle^{2}}.</math>

We can then use the auto correlation function to estimate the [[error bar]]. Luckily, we have a much simpler method based on [[block averaging]].<ref>D. C. Rapaport, ''The Art of Molecular Dynamics Simulation'', Cambridge University Press.</ref>

===Scientific data verification===
{{Main|Scientific misconduct#Exposure of fraudulent data}}
Measurements generally have a small amount of error, and repeated measurements of the same item will generally result in slight differences in readings. These differences can be analyzed, and follow certain known mathematical and statistical properties. Should a set of data appear to be too faithful to the hypothesis, i.e., the amount of error that would normally be in such measurements does not appear, a conclusion can be drawn that the data may have been forged. Error analysis alone is typically not sufficient to prove that data have been falsified or fabricated, but it may provide the supporting evidence necessary to confirm suspicions of misconduct.