TF ErrorAna PropOfErr
Instrumental and Statistical Uncertainties
P=68% = Probability that a measurement of a Gaussian variant will lie within 1 [math]\displaystyle{ \sigma }[/math] of the mean
Example of cosmic counting experiments. Is the variation statistical?
| Date | Time (hrs) | Singles Counts Top (N_1) | Singles Counts Bottom (N_2) | Coincidence Counts | Coinc/Hour | Coinc/min | [math]\displaystyle{ \sqrt{N} }[/math] |
|---|---|---|---|---|---|---|---|
| 11/15 | 21 | 359217 | 383919 | 3581 | 170.52 | 2.84 | |
| 11/5 | 2.0833 | 80713 | 61497 | 310 | 148.8 | 2.48 | |
| 11/2 | 69 | 727769 | 726298 | 10251 | 148.57 |
Taylor Expansion
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]\displaystyle{ 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]\displaystyle{ A = f(L,W) }[/math]
The Taylor series expansion of a function f(x) about the point a is given as
[math]\displaystyle{ f(x) = f(a) + \left . f^{\prime}(x)\right |_{x=a} \frac{x}{1!} + \left . f^{\prime \prime}(x)\right |_{x=a} \frac{x^2}{2!} + ... }[/math]
- [math]\displaystyle{ = \left . \sum_{n=0}^{\infty} f^{(n)}(x)\right |_{x=a} \frac{x^n}{n!} }[/math]
For small values of x (x << 1) we can expand the function about 0 such that
[math]\displaystyle{ \sqrt{1+x} = \left . \sqrt{1-0} \frac{1}{2}(1+x)^{-1/2}\right |_{x=0} \frac{x^1}{1!}+ \left . \frac{1}{2}\frac{-1}{2}(1+x)^{-3/2} \right |_{x=0} \frac{x^2}{2!} }[/math]
- [math]\displaystyle{ =1 + \frac{x}{2} - \frac{x^2}{4} }[/math]
The talylor expansion of a function with two variables[math]\displaystyle{ (x_1 , x_2) }[/math] about the average of the two variables[math]\displaystyle{ (\bar {x_1} , \bar{x_2} ) }[/math] is given by
[math]\displaystyle{ 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]\displaystyle{ 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]
The term
[math]\displaystyle{ f(x_1, x_2)-f(\bar {x}_1, \bar{x}_2) }[/math]
represents a small fluctuation of the function from its average [math]\displaystyle{ f(\bar {x}_1, \bar{x}_2) }[/math] if we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).
Based on the Definition of Variance
- [math]\displaystyle{ \sigma^2 = \frac{\sum_{i=1}^{i=N} (x_i - \bar{x})^2}{N} }[/math]
We can write the variance of the area
- [math]\displaystyle{ \sigma^2_A = \frac{\sum_{i=1}^{i=N} (A_i - \bar{A})^2}{N} }[/math]
- [math]\displaystyle{ = \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]
- [math]\displaystyle{ = \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} \right ] ^2}{N} + \frac{\sum_{i=1}^{i=N} \left [ (W-\bar W) \frac{\partial A}{\partial W} \bigg |_{\bar L \bar W} \right] ^2 }{N} }[/math]
- [math]\displaystyle{ +2 \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \frac{\partial A}{\partial L} \bigg |_{\bar L \bar W} \frac{\partial A}{\partial W} \bigg |_{\bar L \bar W} \right]^2}{N} }[/math]
- [math]\displaystyle{ = \sigma^2_L \left ( \frac{\partial A}{\partial L} \right )^2 +\sigma^2_W \left ( \frac{\partial A}{\partial W} \right )^2 + 2 \sigma^2_{LW} \frac{\partial A}{\partial L} \frac{\partial A}{\partial W} }[/math]
where [math]\displaystyle{ \sigma^2_{LW} = \frac{\sum_{i=1}^{i=N} \left [ (L-\bar{L}) (W-\bar W) \right ]^2}{N} }[/math] is defined as the Covariance between [math]\displaystyle{ L }[/math] and [math]\displaystyle{ W }[/math].
Weighted Mean and variance
If each observable ([math]\displaystyle{ x_i }[/math]) is accompanied by an estimate of the uncertainty in that observable ([math]\displaystyle{ \delta x_i }[/math]) then weighted mean is defined as
- [math]\displaystyle{ \bar{x} = \frac{ \sum_{i=1}^{i=n} \frac{x_i}{\delta x_i}}{\sum_{i=1}^{i=n} \frac{1}{\delta x_i}} }[/math]
The variance of the distribution is defined as
- [math]\displaystyle{ \bar{x} = \sum_{i=1}^{i=n} \frac{1}{\delta x_i} }[/math]