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Compute a moving variance-to-mean ratio (VMR) incrementally.

For a window of size W, the unbiased sample variance is defined as

Equation for the unbiased sample variance.

and the arithmetic mean is defined as

The variance-to-mean ratio (VMR) is thus defined as


npm install @stdlib/stats-incr-mvmr


var incrmvmr = require( '@stdlib/stats-incr-mvmr' );

incrmvmr( window[, mean] )

Returns an accumulator function which incrementally computes a moving variance-to-mean ratio. The window parameter defines the number of values over which to compute the moving variance-to-mean ratio.

var accumulator = incrmvmr( 3 );

If the mean is already known, provide a mean argument.

var accumulator = incrmvmr( 3, 5.0 );

accumulator( [x] )

If provided an input value x, the accumulator function returns an updated accumulated value. If not provided an input value x, the accumulator function returns the current accumulated value.

var accumulator = incrmvmr( 3 );

var F = accumulator();
// returns null

// Fill the window...
F = accumulator( 2.0 ); // [2.0]
// returns 0.0

F = accumulator( 1.0 ); // [2.0, 1.0]
// returns ~0.33

F = accumulator( 3.0 ); // [2.0, 1.0, 3.0]
// returns 0.5

// Window begins sliding...
F = accumulator( 7.0 ); // [1.0, 3.0, 7.0]
// returns ~2.55

F = accumulator( 5.0 ); // [3.0, 7.0, 5.0]
// returns ~0.80

F = accumulator();
// returns ~0.80


  • Input values are not type checked. If provided NaN or a value which, when used in computations, results in NaN, the accumulated value is NaN for at least W-1 future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly before passing the value to the accumulator function.

  • As W values are needed to fill the window buffer, the first W-1 returned values are calculated from smaller sample sizes. Until the window is full, each returned value is calculated from all provided values.

  • The following table summarizes how to interpret the variance-to-mean ratio:

    VMR Description Example Distribution
    0 not dispersed constant
    0 < VMR < 1 under-dispersed binomial
    1 -- Poisson
    >1 over-dispersed geometric, negative-binomial

    Accordingly, one can use the variance-to-mean ratio to assess whether observed data can be modeled as a Poisson process. When observed data is "under-dispersed", observed data may be more regular than as would be the case for a Poisson process. When observed data is "over-dispersed", observed data may contain clusters (i.e., clumped, concentrated data).

  • The variance-to-mean ratio is typically computed on nonnegative values. The measure may lack meaning for data which can assume both positive and negative values.

  • The variance-to-mean ratio is also known as the index of dispersion, dispersion index, coefficient of dispersion, relative variance, and the Fano factor.


var randu = require( '@stdlib/random-base-randu' );
var incrmvmr = require( '@stdlib/stats-incr-mvmr' );

var accumulator;
var v;
var i;

// Initialize an accumulator:
accumulator = incrmvmr( 5 );

// For each simulated datum, update the moving variance-to-mean ratio...
for ( i = 0; i < 100; i++ ) {
    v = randu() * 100.0;
    accumulator( v );
console.log( accumulator() );

See Also


This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.






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