In
statistics, the Fano factor, like the
coefficient of variation, is a measure of the
dispersion
Dispersion may refer to:
Economics and finance
* Dispersion (finance), a measure for the statistical distribution of portfolio returns
* Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variat ...
of a
probability distribution of a
Fano noise. It is named after
Ugo Fano
Ugo Fano (July 28, 1912 – February 13, 2001) was an Italian American physicist, notable for contributions to theoretical physics.
Biography
Ugo Fano was born into a wealthy Jewish family in Turin, Italy. His father was Gino Fano, a professo ...
, an Italian American physicist.
The Fano factor is defined as
:
where
is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and
is the mean of a
random process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
in some time window ''W''. The Fano factor can be viewed as a kind of noise-to-signal ratio; it is a measure of the reliability with which the
random variable could be estimated from a time window that on average contains several
random event
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usua ...
s.
For a
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
, the variance in the count equals the mean count, so ''F'' = 1 (normalization).
If the time window is chosen to be infinity, the Fano factor is similar to the
variance-to-mean ratio
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a ...
(VMR) which in
statistics is also known as the
index of dispersion
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a p ...
.
Use in particle detection
In
particle detector
In experimental and applied particle physics, nuclear physics, and nuclear engineering, a particle detector, also known as a radiation detector, is a device used to detect, track, and/or identify ionizing particles, such as those produced by nu ...
s, the Fano factor results from the energy loss in a collision not being purely statistical. The process giving rise to each individual charge carrier is not independent as the number of ways an atom may be ionized is limited by the discrete electron shells. The net result is a better energy resolution than predicted by purely statistical considerations. For example, if ''w'' is the average energy for a particle to produce a charge carrier in a detector, then the relative
FWHM
In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve me ...
resolution for measuring the particle energy ''E'' is:
:
where the factor of 2.35 relates the standard deviation to the FWHM.
The Fano factor is material specific. Some theoretical values are:
:
Measuring the Fano factor is difficult because many factors contribute to the resolution, but some experimental values are:
:
Use in neuroscience
The Fano factor, along with the coefficient of variation, has been used in
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developme ...
to describe variability in recorded spike trains.
See also
*
Fano noise
*
Index of dispersion
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a p ...
– equivalent to the Fano factor for an infinite time window
*
Ugo Fano
Ugo Fano (July 28, 1912 – February 13, 2001) was an Italian American physicist, notable for contributions to theoretical physics.
Biography
Ugo Fano was born into a wealthy Jewish family in Turin, Italy. His father was Gino Fano, a professo ...
References
{{reflist
Statistical deviation and dispersion
Statistical ratios