Difference between revisions of "TF ErrorAna PropOfErr"
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We can write the variance of the area | We can write the variance of the area | ||
− | :<math>\sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - | + | :<math>\sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - \bar{A})^2}{N}</math> |
− | :<math>= \frac{\sum_{i=1}^{i=N} \left [ (L- | + | :<math>= \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} + (W-\bar W) \frac{\partial A}{\partial W} \bigg |_{\bar L \bar WW} \right] ^2}{N}</math> |
Revision as of 21:23, 9 January 2010
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
The mean that the Area (A) is a function of the Length (L) and the Width (W) of the table.
The Taylor series expansion of a function f(x) about the point a is given as
For small values of x (x << 1) we can expand the function about 0 such that
The talylor expansion of a function with two variables
about the average of the two variables is given by
or
The term
represents a small fluctuation of the function from its average
if we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).Based on the Definition of Variance
We can write the variance of the area