Statistics ==================== Basic Measures ------------------------- The sample distribution has finite size and is what has been measured; the parent distribution is infinite 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: $$\bar{x}= \frac{ \sum_{i=1}^{N} x_i / \sigma_{i}^2}{\sum_{i=1}^{N} 1/\sigma_{i}^2} \pm \sqrt{ \frac{1}{\sum_{i=1}^{N} 1/\sigma_{i}^2}}$$ $\chi^2$ Distribution ------------------------ $\chi^2$ is often written "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.