Mean & Variance

\[ \overline{x}=\frac{\sum_i^nx_i}{n} \] \[ \sigma^2=\frac{1}{N-1}\sum^N_{i=1}(x_i-\overline{x})^2 \] \[ \alpha=\frac{\sigma}{\sqrt{N}} \]

Error Propagation Single Variable Functions

Functional Approach

Symmetric Error: \[ \alpha_Z=|f(\overline{A}+\alpha_A)-f(\overline{A})| \] Asymmetric Error: \[ \alpha_Z^+=|f(\overline{A}+\alpha_A)-f(\overline{A})|\;\;\;\;\;\;\alpha_Z^-=|f(\overline{A})-(\overline{A}-\alpha_A)| \]

Calculus Approach

Note: Always symmetric, valid for small errors only. \[ \alpha_Z=|\frac{df(A)}{dA}|\alpha_A \]

Error Propagation Multi Variable Functions

Functional Approach

Note: MUST NOT USE ON SQUARED VALUES The best estimate of \(Z\) is: \(\overline{Z}=f(\overline A, \overline B)\). \[ \alpha_Z^A=f(\overline A+\alpha_A,\overline B)-f(\overline A, \overline B) \] \[ \alpha_Z^B=f(\overline A,\overline B+\alpha_B)-f(\overline A, \overline B) \] So the Total Error in \(Z\): \[ \alpha_Z=\sqrt{(\alpha_Z^A)^2+(\alpha_Z^B)^2} \]

Calculus Approach

The best estimate of \(Z\) is: \(\overline{Z}=f(\overline A, \overline B)\). \[ \alpha_Z^A=\frac{\partial Z}{\partial A}\cdot\alpha_A \] \[ \alpha_Z^B=\frac{\partial Z}{\partial B}\cdot\alpha_B \] So the Total Error in \(Z\): \[ \alpha_Z=\sqrt{(\alpha_Z^A)^2+(\alpha_Z^B)^2}=\sqrt{(\frac{\partial Z}{\partial A})^2\cdot\alpha_A^2+(\frac{\partial Z}{\partial B})^2\cdot\alpha_B^2} \]

Combining Results

The weight of each result is: \[ w_i=\frac{1}{\alpha^2_i} \] The combined estimate is: \[ x_{CE}=\frac{\sum_iw_ix_i}{\sum_iw_i}=\frac{\sum_i\frac{x_i}{\alpha^2_i}}{\sum_i\frac{1}{\alpha^2_i}} \] The inverse of the square of the standard error of the weighted mean is the sum of the weightings: \[ \frac{1}{\alpha^2_{CE}}=\sum_i\frac{1}{\alpha^2_i} \]

Chauvenet’s Criterion

Chauvenet's criterion: a data point is rejected from a sample if the number of events we expect to be further from the mean than the suspect point, for the sample's mean and standard deviation, is less than half.

1. Calculate the mean and standard deviation.
2. Find the difference between the mean and the potential outlier
3. Find the probability that a result would randomly differ from the mean by the same amount
4. Multiply with the number of data points
5. If the outlier is rejected and the mean and standard deviation are recalculated for the remaining data points

Poisson Distribution

The average count \(\overline{N}\), is given by: \[ \overline{N}=\mu\cdot t \] where \(\mu\) is the average count rate, and \(t\) is the counting time \[ \textrm{Standard Deviation: }\sigma=\sqrt{\overline{N}} \] \[ \textrm{Results: }(\overline{N}\pm\sigma) \]

Approximating at Large Means

Poisson: \[ P(N;\overline{N})=\frac{N^{-N}}{N!}e^{-\overline{N}} \] Gaussian: \[ G(x;\overline{x},\sigma)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\overline{x})^2}{2\sigma^2}}=\frac{1}{\sqrt{2\pi\overline{N}}}e^{-\frac{(N-\overline{N})^2}{2\overline{N}}} \]

Binomial Distribution

If the probability of an event happening (success) is \(p\), and probability of event not taking place (failure) is \(1-p=q\), the probability \(P(r)\) for a total of \(r\) successes in \(N\) attempts is: \[ \boxed{P(r)=\frac{N!}{r!(N-r)!}p^r(1-p)^{N-r}} \]

Method of least squares

The method of least squares can be used to find the best-fit. For a set of points \((x_i,y_i)\):