The variance is defined as $$ {\rm var}(x) = {1\over N-1} \sum_{i=1}^N (x_i - \bar x )^2 $$

If x is an array, compute the variance along the first non-singleton dimensions of x.

The optional argument w determines the weighting scheme to use. Valid values are:

0 [default]:

Normalize with $N-1$ (population variance). This provides the square root of the best unbiased estimator of the variance.

Normalize with $N$ (sample variance). This provides the square root of the second moment around the mean.

Compute the weighted variance with non-negative weights. The length of w must equal the size of x in the operating dimension. NaN values are permitted in w, will be multiplied with the associated values in x, and can be excluded by the nanflag option.

Similar to vector weights, but w must be the same size as x. If the operating dimension is supplied as vecdim or "all" and w is not a scalar, w must be an same-sized array.

Note: w must always be specified before specifying any of the following dimension options. To use the default value for w you may pass an empty input argument [].

The optional variable dim forces var to operate over the specified dimension, which must be a positive integer-valued number. Specifying any singleton dimension in x, including any dimension exceeding ndims (x), will result in a variance of 0.

Specifying the dimensions as vecdim, a vector of non-repeating dimensions, will return the variance calculated over the array slice defined by vecdim. If vecdim indexes all dimensions of x, then it is equivalent to the option "all". Any dimension in vecdim greater than ndims (x) is ignored.

Specifying the dimension as "all" will force var to operate on all elements of x, and is equivalent to var (x(:)).

The optional variable nanflag specifies whether to include or exclude NaN values from the calculation using any of the previously specified input argument combinations. The default value for nanflag is "includenan" which keeps NaN values in the calculation. To exclude NaN values set the value of nanflag to "omitnan". The output will still contain NaN values if x consists of all NaN values in the operating dimension.

The optional second output variable m contains the mean of the elements of x used to calculate the variance. If v is the weighted variance, then m is also the weighted mean.

See also: std, mean, cov, skewness, kurtosis, moment

Source Code: var