Editing Propagation of uncertainty (section)

==Example formulae==
This table shows the variances and standard deviations of simple functions of the real variables <math>A, B</math> with standard deviations <math>\sigma_A, \sigma_B,</math> [[Covariance and correlation|covariance]] <math>\sigma_{AB} = \rho_{AB} \sigma_A \sigma_B,</math> and correlation <math>\rho_{AB}.</math> The real-valued coefficients <math>a</math> and <math>b</math> are assumed exactly known (deterministic), i.e., <math>\sigma_a = \sigma_b = 0.</math>

In the right-hand columns of the table, <math>A</math> and <math>B</math> are [[expected value|expectation values]], and <math>f</math> is the value of the function calculated at those values.

{| class="wikitable"
! Function !! Variance !! Standard deviation
|-
| <math>f = aA\,</math>
| <math>\sigma_f^2 = a^2\sigma_A^2</math>
| <math>\sigma_f = |a|\sigma_A</math>
|-
| <math>f = A + B</math>
| <math>\sigma_f^2 = \sigma_A^2 + \sigma_B^2 + 2\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{\sigma_A^2 + \sigma_B^2 + 2\sigma_{AB}}</math>
|-
| <math>f = A - B</math>
| <math>\sigma_f^2 = \sigma_A^2 + \sigma_B^2 - 2\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{\sigma_A^2 + \sigma_B^2 - 2\sigma_{AB}}</math>
|-
| <math>f = aA + bB</math>
| <math>\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 + 2ab\,\sigma_{AB}}</math>
|-
| <math>f = aA - bB</math>
| <math>\sigma_f^2 = a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}</math>
| <math>\sigma_f = \sqrt{a^2\sigma_A^2 + b^2\sigma_B^2 - 2ab\,\sigma_{AB}}</math>
|-
| <math>f = AB</math>
| <math>\sigma_f^2 \approx f^2 \left[\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 + 2\frac{\sigma_{AB}}{AB} \right]</math><ref>{{cite web |url=http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_Propagation_sp13.pdf |title=A Summary of Error Propagation |page=2 |access-date=2016-04-04 |archive-url = https://web.archive.org/web/20161213135602/http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_Propagation_sp13.pdf |archive-date=2016-12-13 |url-status=dead }}</ref><ref>{{cite web |url=http://web.mit.edu/fluids-modules/www/exper_techniques/2.Propagation_of_Uncertaint.pdf |title=Propagation of Uncertainty through Mathematical Operations |page=5 |access-date=2016-04-04}}</ref>
| <math>\sigma_f \approx \left| f \right| \sqrt{ \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 + 2\frac{\sigma_{AB}}{AB} }</math>
|-
| <math>f = \frac{A}{B}</math>
| <math>\sigma_f^2 \approx f^2 \left[\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} \right]</math><ref>{{cite web |url=http://www.sagepub.com/upm-data/6427_Chapter_4__Lee_%28Analyzing%29_I_PDF_6.pdf |title=Strategies for Variance Estimation |page=37 |access-date=2013-01-18}}</ref>
| <math>\sigma_f \approx \left| f \right| \sqrt{ \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} }</math>
|-
| <math>f = \frac{A}{A+B}</math>
| <math>\sigma_f^2 \approx \frac{f^2}{\left(A+B\right)^2} \left(\frac{B^2}{A^2}\sigma_A^2  +\sigma_B^2 - 2\frac{B}{A} \sigma_{AB} \right)</math>
| <math>\sigma_f \approx \left|\frac{f}{A+B}\right| \sqrt{\frac{B^2}{A^2}\sigma_A^2  +\sigma_B^2 - 2\frac{B}{A} \sigma_{AB}  }</math>
|-
| <math>f = a A^b</math>
| <math>\sigma_f^2 \approx \left( {a}{b}{A}^{b-1}{\sigma_A} \right)^2 = \left( \frac{{f}{b}{\sigma_A}}{A} \right)^2 </math>
| <math>\sigma_f \approx \left| {a}{b}{A}^{b-1}{\sigma_A} \right| = \left| \frac{{f}{b}{\sigma_A}}{A} \right| </math>
|-
| <math>f = a \ln(bA)</math>
| <math>\sigma_f^2 \approx \left(a \frac{\sigma_A}{A} \right)^2</math><ref name=harris2003>{{citation |first1=Daniel C. |last1=Harris | title=Quantitative chemical analysis |edition=6th |publisher=Macmillan |year=2003 |isbn=978-0-7167-4464-1 |page=56 |url=https://books.google.com/books?id=csTsQr-v0d0C&pg=PA56 }}</ref>
| <math>\sigma_f \approx \left| a \frac{\sigma_A}{A}\right|</math>
|-
| <math>f = a \log_{10}(bA)</math>
| <math>\sigma_f^2 \approx \left(a \frac{\sigma_A}{A \ln(10)} \right)^2</math><ref name=harris2003/>
| <math>\sigma_f \approx \left| a \frac{\sigma_A}{A \ln(10)} \right|</math>
|-
| <math>f = a e^{bA}</math>
| <math>\sigma_f^2 \approx f^2 \left( b\sigma_A \right)^2</math><ref>{{cite web|url=http://www.foothill.edu/psme/daley/tutorials_files/10.%20Error%20Propagation.pdf|date=October 9, 2009|title=Error Propagation tutorial|work=Foothill College | access-date=2012-03-01}}</ref>
| <math>\sigma_f \approx \left| f \right| \left| \left( b\sigma_A \right) \right| </math>
|-
| <math>f = a^{bA}</math>
| <math>\sigma_f^2 \approx f^2 (b\ln(a)\sigma_A)^2</math>
| <math>\sigma_f \approx \left| f \right| \left| b \ln(a) \sigma_A \right|</math>
|-
| <math>f = a \sin(bA)</math>
| <math>\sigma_f^2 \approx \left[ a b \cos(b A) \sigma_A \right]^2</math>
| <math>\sigma_f \approx \left| a b \cos(b A) \sigma_A \right|</math>
|-
| <math>f = a \cos \left( b A \right)\,</math>
| <math>\sigma_f^2 \approx \left[ a b \sin(b A) \sigma_A \right]^2</math>
| <math>\sigma_f \approx \left| a b \sin(b A) \sigma_A \right|</math>
|-
|<math>f = a \tan \left( b A \right)\,</math>
|<math>\sigma_f^2 \approx \left[ a b \sec^2(b A) \sigma_A \right]^2</math>
|<math>\sigma_f \approx \left| a b \sec^2(b A) \sigma_A \right|</math>
|-
| <math>f = A^B</math>
| <math>\sigma_f^2 \approx f^2 \left[ \left( \frac{B}{A}\sigma_A \right)^2 +\left( \ln(A)\sigma_B \right)^2 + 2 \frac{B \ln(A)}{A} \sigma_{AB} \right]</math>
| <math>\sigma_f \approx \left| f \right| \sqrt{ \left( \frac{B}{A}\sigma_A \right)^2 +\left( \ln(A)\sigma_B \right)^2 + 2 \frac{B \ln(A)}{A} \sigma_{AB} } </math>
|-
| <math>f = \sqrt{aA^2 \pm bB^2}</math>
| <math>\sigma_f^2 \approx \left(\frac{A}{f}\right)^2 a^2\sigma_A^2 + \left(\frac{B}{f}\right)^2 b^2\sigma_B^2 \pm 2ab\frac{AB}{f^2}\,\sigma_{AB}</math>
| <math>\sigma_f \approx \sqrt{\left(\frac{A}{f}\right)^2 a^2\sigma_A^2 + \left(\frac{B}{f}\right)^2 b^2\sigma_B^2 \pm 2ab\frac{AB}{f^2}\,\sigma_{AB}}</math>
|}

For uncorrelated variables (<math>\rho_{AB} = 0</math>, <math>\sigma_{AB} = 0</math>) expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation, gives
<math display="block">f = ABC; \qquad \left(\frac{\sigma_f}{f}\right)^2 \approx \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2+ \left(\frac{\sigma_C}{C}\right)^2.</math>

For the case <math>f = AB </math> we also have Goodman's expression<ref name="Goodman1960"/> for the exact variance: for the uncorrelated case it is
<math display="block">\operatorname{V}[XY] = \operatorname{E}[X]^2 \operatorname{V}[Y] + \operatorname{E}[Y]^2 \operatorname{V}[X] + \operatorname{E}\left[\left(X - \operatorname{E}(X)\right)^2 \left(Y - \operatorname{E}(Y)\right)^2\right],</math>
and therefore we have
<math display="block">\sigma_f^2 = A^2\sigma_B^2 + B^2\sigma_A^2 +  \sigma_A^2\sigma_B^2.</math>

===Effect of correlation on differences===
If ''A'' and ''B'' are uncorrelated, their difference ''A'' − ''B'' will have more variance than either of them. An increasing positive correlation (<math>\rho_{AB} \to 1</math>) will decrease the variance of the difference, converging to zero variance for perfectly correlated variables with the [[homoscedastic|same variance]]. On the other hand, a negative correlation (<math>\rho_{AB} \to -1</math>) will further increase the variance of the difference, compared to the uncorrelated case.

For example, the self-subtraction ''f'' = ''A'' − ''A'' has zero variance <math>\sigma_f^2 = 0</math> only if the variate is perfectly [[autocorrelation|autocorrelated]] (<math>\rho_A = 1</math>). If ''A'' is uncorrelated, <math>\rho_A = 0,</math> then the output variance is twice the input variance, <math>\sigma_f^2 = 2\sigma^2_A.</math> And if ''A'' is perfectly anticorrelated, <math>\rho_A = -1,</math> then the input variance is quadrupled in the output, <math>\sigma_f^2 = 4 \sigma^2_A</math> (notice <math>1 - \rho_A = 2</math> for ''f'' = ''aA'' − ''aA'' in the table above).