Editing Propagation of uncertainty (section)

=== Simplification ===
Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:<ref>{{cite journal|last=Ku|first=H. H.|date=October 1966 |title=Notes on the use of propagation of error formulas|url=http://nistdigitalarchives.contentdm.oclc.org/cdm/compoundobject/collection/p16009coll6/id/99848/rec/1|journal=Journal of Research of the National Bureau of Standards | volume=70C|issue=4|page=262|doi=10.6028/jres.070c.025|issn=0022-4316|access-date=3 October 2012|doi-access=free}}</ref>
<math display="block">s_f = \sqrt{ \left(\frac{\partial f}{\partial x}\right)^2 s_x^2 + \left(\frac{\partial f}{\partial y} \right)^2 s_y^2 + \left(\frac{\partial f}{\partial z} \right)^2 s_z^2 + \cdots}</math>
where <math>s_f</math> represents the standard deviation of the function <math>f</math>, <math>s_x</math> represents the standard deviation of <math>x</math>, <math>s_y</math> represents the standard deviation of <math>y</math>, and so forth.

This formula is based on the linear characteristics of the gradient of <math>f</math> and therefore it is a good estimation for the standard deviation of <math>f</math> as long as <math>s_x, s_y, s_z,\ldots</math> are small enough. Specifically, the linear approximation of <math>f</math> has to be close to <math>f</math> inside a neighbourhood of radius <math>s_x, s_y, s_z,\ldots</math>.<ref>{{Cite book |last=Clifford |first=A. A. |title=Multivariate error analysis: a handbook of error propagation and calculation in many-parameter systems |publisher=John Wiley & Sons |year=1973 |isbn=978-0470160558 }}{{page needed|date=October 2012}}</ref>