In statistics, the Fano factor, like the

coefficient of variation In probability theory and statistics, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed ...

, 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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...

of a

Fano noise Fano noise is a fluctuation of an electric charge obtained in a detector (in spite of constant value of the measured quantity, which is usually an energy), arising from processes in the detector. It was first described by Ugo Fano in 1947, as a flu ...

. 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 profes ...

, an Italian American physicist. The Fano factor is defined as :

F=\frac,

where

\sigma_W^2

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 number ...

and

\mu_W

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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

could be estimated from a time window that on average contains several random events. 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 pro ...

(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 pr ...

Use in particle detection

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 ...

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 resolution for measuring the particle energy ''E'' is: :

R = \frac = 2.35 \sqrt,

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 science, scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a Multidisciplinary approach, multidisciplinary science that combines physiology, an ...

to describe variability in recorded spike trains.

References

{{reflist Statistical deviation and dispersion Statistical ratios

Use in particle detection

Use in neuroscience

See also

References