<aside> 💡 PRACTICE: Plotting graphs quickly. SAO 2018 Specimen is quite good
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Quadrature:
$$ \Delta Z^2=\left( \frac{\partial Z}{\partial X}\Delta X\right)^2+\left( \frac{\partial Z}{\partial Y}\Delta Y\right)^2 $$
Simplifications:
$$ f=x+y \implies \Delta f=\Delta x+\Delta y\\f=x^ay^b \implies \frac{\Delta f}{f}=\sqrt{\left( a\frac{\Delta x}{x}\right)^2+\left( b\frac{\Delta y}{y}\right)^2} $$
Variance: $\sigma^2$
Once the standard deviation is computed, error estimation of the mean value is $\delta \bar{x} = \sigma / \sqrt{N}$
If each value also has a certain uncertainty associated with it, there is a different formula to get the mean (weighted mean):
$$ \bar x=\frac{\sum^N_{i=1}x_i/\delta x_i^2}{\sum^N_{i=1}1/\delta x_i^2} \qquad \delta \bar x=\frac{1}{\sqrt{\sum^N_{i=1}1/\delta x^2}} $$