One way we can tell we have an observational error is to perform a measurement many times. Since in theory we should be getting the same result each time, if we get different results each time we can attribute that difference to observational error. This means the result of an experiment is usually not a single number but a distribution of numbers.
Mathematically we can describe the distribution of numbers with a probability distribution function or pdf. This is a function that gives the probability of getting different numbers in a distribution. There are a number of distribution functions that are quite famous in statistics.
- Binomial Distribution – this is for the case where you want the number of successes is a sequence of repeatable experiments where there are 2 possible outcomes. The probability of getting k successes in n tries where the probability each experiment is p is give by
$ P(k; n, p) = {n! \over {k!( n-k)!}} p^k (1-p)^{n-k}$
- Poisson Distribution – this distribution describes the situation when discrete independent events are occurring in time with an average event rate of $\mu$. Then the probability of k events in that time frame is given by
$P(k) = {\mu^k \over{k!}} e^{-\mu}$
- Gaussian or Normal Distribution – this is the distribution that according the the central limit theorem a large number of independent random occurrences added together will approach. Thus both the binomial and Poisson distribution can be approximated by the normal distribution when large. The Gaussian distribution is described by its mean, $\mu$, and standard deviation, $\sigma$.
$P(x) = {1 \over {2 \pi \sigma}}\exp^{-{(x – \mu)^2 \over {2 \sigma^2}}}$
This site offers a nice visual guide to probability density functions. .
Readings:
think-stats: modeling distributions
