Greenwood statistic

In general, for a given sequence of events in time or space the statistic is given by:.[1]

${\displaystyle G(n)=\sum _{i=1}^{n+1}D_{i}^{2},}$

where ${\displaystyle D_{i}}$ represents the interval between events or points in space and is a number between 0 and 1 such that the sum of all ${\displaystyle D_{i}=1}$.

Where intervals are given by numbers that do not represent a fraction of the time period or distance, the Greenwood statistic is modified [2] and is given by:

${\displaystyle G(n)={\frac {\sum _{i=1}^{n+1}X_{i}^{2}}{T_{n}^{2}}},}$

where:

${\displaystyle T_{n}=\sum _{i=1}^{n+1}X_{i},}$

and ${\displaystyle X_{i}}$ represents the length of the 'ith interval, which is either the time between events or the distances between points in space.

A reformulation of the statistic yields

${\displaystyle G(n)={\tfrac {1}{n+1}}({\tfrac {n}{n+1}}C_{v}^{2}+1),}$

where ${\displaystyle C_{v}}$ is the sample coefficient of variation of the n + 1 interval lengths.

The Greenwood statistic is a comparative measure that has a range of values between 0 and 1. For example, applying the Greenwood statistic to the arrival of 11 buses in a given time period of say 1 hour, where in the first example all eleven buses arrived at a given point each 6 minutes apart, would give a result of roughly 0.10. However, in the second example if the buses became bunched up or clustered so that 6 buses arrived 10 minutes apart and then 5 buses arrived 2 minutes apart in the last 10 minutes, the result is roughly 0.17. The result for a random distribution of 11 bus arrival times in an hour will fall somewhere between 0.10 and 0.17. So this can be used to tell how well a bus system is running and in a similar way, the Greenwood statistic was also used to determine how and where genes are placed in the chromosomes of living organisms.[3] This research showed that there is a definite order to where genes are placed, particularly with regard to what function the genes perform, and this is important in the science of genetics.