A quantity which is calculated using quantities with known uncertainties will have an uncertainty based upon the uncertainty of the quantities used in the calculation.
To determine the uncertainty in a quantity which is a function of other quantities, you can consider the dependence of these quantities in terms of a tayler expansion
Consider a calculation of a Table's Area
[math]A= L \times W[/math]
The mean that the Area (A) is a function of the Length (L) and the Width (W) of the table.
[math]A = f(L,W)[/math]
The Taylor series expansion of a function f(x) about the point a is given as
[math]f(x) = f(a) + f^{\prime}(x)|_{x=a} \frac{x}{1!} + f^{\prime \prime}(x)|_{x=a} \frac{x^2}{2!} + ...[/math]
- [math]= \sum_{n=0}^{infty} f^{(n)}(x)|_{x=a} \frac{x^n}{n!}[/math]
For small values of x (x << 1) we can expand the function about 0 such that
[math]\sqrt{1+x} = \sqrt{1-0} | \frac{1}{2}(1+x)^{-1/2}|_{x=0} \frac{x^1}{1!}+ \frac{1}{2}\frac{-1}{2}(1+x)^{-3/2}|_{x=0} \frac{x^2}{2!}[/math]
- [math]=1 + \frac{x}{2} - \frac{x^2}{4}[/math]
The talylor expansion of a function with two variables[math] (x_1 , x_1)[/math] about the average of the two variables[math] (\bar {x_1} , \bar{x_2} )[/math] is given by
[math]f(x_1, x_2)=f(\bar {x_1}, \bar{x_2})+(x_1-\bar {x_1}) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = \bar {x_1}, x_2 = \bar{x_2})} +(x_2-{\bar{x_2}) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = \bar {x_1}, x_2 = \bar{x_2})}[/math]
or
[math]f(x_1, x_2)-f(\bar {x_1}, {x_o}_2)=(x_1-\bar {x_1}) \frac{\partial f}{\partial x_1}\bigg |_{(x_1 = \bar {x_1}, x_2 = x_{02})} +(x_2-{x_o}_2) \frac{\partial f}{\partial x_2}\bigg |_{(x_1 = \bar {x_1}, x_2 = x_{02})}[/math]
The term
[math]f(x_1, x_2)-f({x_o}_1, {x_o}_2)[/math]
represents a small fluctuation of the function from its average [math]f({x_o}_1, {x_o}_2)[/math] ff we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).
Based on the Definition of Variance
- [math]\sigma^2 = \frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N}[/math]
We can write the variance of the area
- [math]\sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - A_0)^2}{N}[/math]
- = \\frac{\sum_{i=1}^{i=N} \left [ (L-L_0) \frac{\partial A}{\partial L} \bigg |_{L_0,W_0} + (W-W_0) \frac{\partial A}{\partial W} \bigg |_{L_0,W_0} \right] ^2</math>
[1] Forest_Error_Analysis_for_the_Physical_Sciences