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 diff --git a/math/statistics.page b/math/statistics.pagenew file mode 100644index 0000000..446450f--- /dev/null+++ b/math/statistics.page@@ -0,0 +1,80 @@+Statistics+====================++Basic Measures+-------------------------+The sample distribution has finite size and is what has been measured; the+parent distribution is inifinite and smooth and is the limit case of the+sample distribution.++The mean, or average, is (of course):+$$\langle x \rangle = \frac{1}{N} \sum_{i=1}^{N}x_i$$++The variance is;+$$s^{2}_x = \frac{1}{N-1}\sum^{N}_{i=1}\left(x-\langle x \rangle\right)^2$$++The standard deviation is the square root of the variance; the standard +deviation of the parent distribution is represented by $\sigma_x$ instead of+$s_x$. The mean of the parent distribution is $\mu$ instead of $\bar{x}$.++Binomial Distribution+-------------------------+If we are playing a yes/no game (eg flipping a coin), the binomial distribution+represents the probability of getting 'yes' $x$ times out of $n$ if $p$ is the+probability of getting 'yes' for a single attempt.++$$P(x;n,p) = \frac{n!}{x! (n-x)!} p^x (1-p)^{n-x}$$++The mean of this distribution is $\mu = np$, and $\sigma$ = \sqrt{np (1-p)}.++Poisson Distribution+------------------------++$$P(x,\mu) = \frac{\mu^x}{x!} e^{-\mu}$$++The mean is $\mu$, and $\sigma=\sqrt{\mu}$.++Gaussian Distribution+--------------------------+The classic! Also called a normal distribution.++$$P(x;\mu,\sigma) = \frac{1}{2\pi \sigma} e^{-\left(\frac{(x-\mu)^2}{2\sigma^2}\right)}$$++The mean is $\mu$ and the deviation is $\sigma=\sqrt(\mu)$.++Lorentzian Distribution+---------------------------+This distribution represents damped resonance; it is also the Fourier +transform of an exponentially decaying sinusoid.++$$P(x;\mu,\Gamma) = \frac{1}{\pi} \frac{\Gamma/2}{(x-\mu)^2 + (\Gamma/2)^2}$$++where the mean is $\mu$ and the linewidth (the width of the peak) is $\Gamma$.++Error Analysis+-------------------+For a given measurement, the error on the mean is not the standard deviation +(which is a measure of the statistics), it is $\frac{s_x}{\sqrt{N}}$: the+standard deviation should stay roughly constant as $N$ gets very large, but+the error on the mean should get smaller. More elaborately, if the errors are+different for each individual measurement, the mean will be:++$$\bar{x}=+ \frac{ \sum_{i=1}^{N} x_i / \simga_{i}^2}{\sum_{i=1}^{N} 1/\simga_{i}^2}+ \pm \sqrt{ \frac{1}{\sum_{i=1}^{N} 1/\simga_{i}^2}}$$++$\Chi^2$ Distribution+------------------------+$\Chi^2$ is often writen "chi-squared" and is a metric for how well a fit +curve matches uncertain data.++$$\Chi^2 = \sum_{i=1}^{N}\left(\frac{x_i-\mu_i}{\sigma{i}}\right)^2$$++The number of degrees of freedom of the system is the number of measurements+$N$ minus the number of variable parameters in a curve fit $N_c$: $\nu = N-N_c$.++The reduced $\Chi^2$ value is $\Chi^{2}_r = \Chi^2 /\nu$. You want $\Chi^{2}_r$+to be around (but not exactly!) 1; if it is significantly larger there are +probably too many degrees of freedom, while if significantly smaller the fit is+bad. +