# Bunching parameter

In statistics as applied in particular in particle physics, when fluctuations of some observables are measured, it is convenient to transform the multiplicity distribution to the bunching parameters:

${\displaystyle \eta _{q}={\frac {q}{q-1}}{\frac {P_{q}P_{q-2}}{P_{q-1}^{2}}},}$

where ${\displaystyle P_{n}}$ is probability of observing ${\displaystyle n}$ objects inside of some phase space regions. The bunching parameters measure deviations of the multiplicity distribution ${\displaystyle P_{n}}$ from a Poisson distribution, since for this distribution

${\displaystyle \eta _{q}=1}$.

Uncorrelated particle production leads to the Poisson statistics, thus deviations of the bunching parameters from the Poisson values mean correlations between particles and dynamical fluctuations.

Normalised factorial moments have also similar properties. They are defined as

${\displaystyle F_{q}=\langle n\rangle ^{-q}\sum _{n=q}^{\infty }{\frac {n!}{(n-q)!}}P_{n}.}$

## Numeric implementation

Bunching parameters and normalized factorial moments are included to the DataMelt program for data analysis and scientific computing.