variance

Arguments

x

real or complex vector or matrix. A hypermatrix is accepted only for undirectional computations variance(x) or variance(x,"*",m)

orien

the orientation of the computation. Valid values are

• 1 or "r": result is a row, after a column-wise computation.
• 2 or "c": result is a column, after a row-wise computation.
• "*": full undirectional computation (default); explicitly required when m is used.

m

The known mean of the underlying statistical distribution law (assuming that it is known).

• "*" mode (default): m must be scalar
• "r" or 1 mode: m is a row of length size(x,2). The variance along the column #j is computed using m(j) as the mean for the considered column. If m(j) is the same for all columns, it can be provided as a scalar m.
• "c" or 2 mode: m is a column of length size(x,1). The variance along the row #i is computed using m(i) as the mean for the considered row. If m(i) is the same for all rows, it can be provided as a scalar m.

When m is not provided, the variance is built dividing the quadratic distance of n values to mean(x) (or mean(x,"c") or mean(x,"r")) by n-1 (n being length(x) or size(x,1) or size(x,2)). If the elements of x are mutually independent, the result is then statistically unbiased.

Else, the variance is built dividing the quadratic distance of values to m by the number n of considered values (n being length(x) or size(x,1) or size(x,2)).

If a true value m independent from x elements is used, x and m values are mutually independent, and the result is then unbiased.

When the special value m = %nan is provided, the variance is still normalized by n (not n-1) but is computed using m=mean(x) instead (or m = mean(x,"c") or m = mean(x,"r")). This m does not bring independent information, and yields a statistically biased result.

s

The variance of unweighted values of x elements. It is a scalar or a column vector or a row vector according to orien.

mc

Scalar or orien-wise mean of x elements (unweighted) (= mean(x,..)), as computed before and used as reference in the variance.

Description

This function computes the variance of the real or complex numbers stored into a vector or matrix x. If x is complex, variance(x,..) = variance(real(x),..) + variance(imag(x),..) is returned.

For a vector, a matrix, or a hypermatrix x, s = variance(x) returns in the scalar s the variance of all the entries of x.

s = variance(x,"c") (or, equivalently, s = variance(x,2)) is the columnwise variance: s is a column vector, with s(j) = variance(x(j,:)).

s = variance(x,"r") (or, equivalently, s = variance(x,1)) is the rowwise variance: s is a row vector, with s(i) = variance(x(:,i)).

The second output argument m is the mean of the input, with respect to the orien parameter.

The variance(x, "*"|"c"|"r", 1) synopsis used only in Scilab 5.4.1 must be replaced with variance(x, "*"|"c"|"r", %nan). variance(x, "*"|"c"|"r", 1) will warn the user until Scilab 6.0. Indeed, 1 will be now considered as m=1. If 1 is the true value provided as m, the warning may be avoided entering 1+%eps instead of 1.

Examples

x = [ 0.2113249 0.0002211 0.6653811; 0.7560439 0.4453586 0.6283918 ]
s = variance(x)
s = variance(x, "r")
s = variance(x, "c")

// The underlying statistical distribution and its mean are known:
x = grand(100, 5, "unf", 0, 7);      // Uniform distribution on [0, 7]
// => the true asymptotic mean is (0+7)/2 = 3.5 and variance = (7-0)^2/12
(7-0)^2/12                  // True asymptotic variance
s = variance(x)             // Unbiased (division by n-1).
s = variance(x, "*", 3.5)   // Unbiased (division by n). Always >= variance(x)
s = variance(x, "*", %nan)    // Biased   (division by n). Always <= variance(x)
// Across columns:
s = variance(x, "c")
s = variance(x, "c", 3.5)
s = variance(x, "c", %nan)

// With complex numbers uniformly distributed on [0,1] + [0,1].i:
x = rand(4, 3) + rand(4, 3)*%i
s = variance(x)
s = variance(x, "*", 0.5 + 0.5*%i)
s = variance(x, "*", %nan)
s = variance(x, "r")
s = variance(x, "c")

// With a hypermatrix
x = rand(3, 2, 2)    // Uniform distribution on [0, 1]
s = variance(x)
s = variance(x, "*", 0.5)
s = variance(x, "*", %nan)
// s = variance(x, "r")  // Is not supported for a hypermatrix
// s = variance(x, "c")  // Is not supported for a hypermatrix

See also

• variancef — variance (and mean) of a vector or matrix of frequency-weighted real or complex numbers
• mtlb_var — Matlab var emulation function
• stdev — standard deviation (row orcolumn-wise) of vector/matrix entries

Bibliography

Wonacott, T.H. & Wonacott, R.J.; Introductory Statistics, fifth edition, J.Wiley & Sons, 1990.

History