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variance
variance (and mean) of a vector or matrix (or hypermatrix) of real or complex numbers
Syntax
[s [,mc]] = variance(x [,orien [,m]]) [s, mc] = variance(x) [s, mc] = variance(x, "r"1 ) [s, mc] = variance(x, "c"2 ) [s, mc] = variance(x, "*" , %nan) [s, mc] = variance(x, "r"1, %nan) [s, mc] = variance(x, "c"2, %nan) s = variance(x, "*", m) s = variance(x, "r"1, m) s = variance(x, "c"2, m)
Arguments
 x
real or complex vector or matrix. A hypermatrix is accepted only for undirectional computations
variance(x)
orvariance(x,"*",m)
 orien
the orientation of the computation. Valid values are
 1 or "r": result is a row, after a columnwise computation.
 2 or "c": result is a column, after a rowwise 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 lengthsize(x,2)
. The variance along the column #j is computed usingm(j)
as the mean for the considered column. Ifm(j)
is the same for all columns, it can be provided as a scalarm
.  "c" or 2 mode:
m
is a column of lengthsize(x,1)
. The variance along the row #i is computed usingm(i)
as the mean for the considered row. Ifm(i)
is the same for all rows, it can be provided as a scalarm
.
When
m
is not provided, thevariance
is built dividing the quadratic distance ofn
values tomean(x)
(ormean(x,"c")
ormean(x,"r")
) byn1
(n
beinglength(x)
orsize(x,1)
orsize(x,2)
). If the elements ofx
are mutually independent, the result is then statistically unbiased.Else, the
variance
is built dividing the quadratic distance of values tom
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
andm
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 n1) but is computed usingm=mean(x)
instead (orm = mean(x,"c")
orm = mean(x,"r")
). Thism
does not bring independent information, and yields a statistically biased result. "*" mode (default):
 s
 The variance of unweighted values of
x
elements. It is a scalar or a column vector or a row vector according toorien
.  mc
 Scalar or
orien
wise mean ofx
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 = (70)^2/12 (70)^2/12 // True asymptotic variance s = variance(x) // Unbiased (division by n1). 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
Bibliography
Wonacott, T.H. & Wonacott, R.J.; Introductory Statistics, fifth edition, J.Wiley & Sons, 1990.
History
Version  Description 
5.5.0 

5.4.1 

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