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Variance
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Variance
V = var(A) example
V = var( ___ ,w) example
V = var( ___ ,dim) example
V = var( ___ ,nanflag) example
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var
Syntax
DescriptionV = var(A) returns the variance of the elements of A along the first array dimension whosesize does not equal 1.
If A is a vector of observations, the variance is a scalar.
If A is a matrix whose columns are random variables and whose rows are observations, V isa row vector containing the variances corresponding to each column.
If A is a multidimensional array, then var(A) treats the values along the first arraydimension whose size does not equal 1 as vectors. The size of this dimension becomes 1while the sizes of all other dimensions remain the same.
The variance is normalized by the number of observations‐1 by default.
If A is a scalar, var(A) returns 0. If A is a 0by0 empty array, var(A) returns NaN.
V = var( ___ ,w) specifies a weighting scheme for any of the previous syntaxes. When w = 0(default), V is normalized by the number of observations‐1. When w = 1, it is normalized by thenumber of observations. w can also be a weight vector containing nonnegative elements. In thiscase, the length of w must equal the length of the dimension over which var is operating.
V = var( ___ ,dim) returns the variance along the dimension dim for any of the previoussyntaxes. To maintain the default normalization while specifying the dimension of operation, setw = 0 in the second argument.
V = var( ___ ,nanflag) specifies whether to include or omit NaN values from the calculation forany of the previous syntaxes. For example, var(A,'includenan') includes all NaN values in Awhile var(A,'omitnan') ignores them.
Examples
Variance of Matrix
Create a matrix and compute its variance.
A = [4 ‐7 3; 1 4 ‐2; 10 7 9]; var(A)
ans =
21.0000 54.3333 30.3333
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Variance of Array
Create a 3D array and compute its variance.
A(:,:,1) = [1 3; 8 4]; A(:,:,2) = [3 ‐4; 1 2]; var(A)
ans(:,:,1) =
24.5000 0.5000
ans(:,:,2) =
2 18
Specify Variance Weight Vector
Create a matrix and compute its variance according to a weight vector w.
A = [5 ‐4 6; 2 3 9; ‐1 1 2]; w = [0.5 0.25 0.25]; var(A,w)
ans =
6.1875 9.5000 6.1875
Specify Dimension for Variance
Create a matrix and compute its variance along the first dimension.
A = [4 ‐2 1; 9 5 7]; var(A,0,1)
ans =
12.5000 24.5000 18.0000
Compute the variance of A along the second dimension.
var(A,0,2)
ans =
9 4
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Variance Excluding NaN
Create a vector and compute its variance, excluding NaN values.
A = [1.77 ‐0.005 3.98 ‐2.95 NaN 0.34 NaN 0.19]; V = var(A,'omitnan')
V =
5.1970
Input Arguments
A — Input arrayvector | matrix | multidimensional array
Input array, specified as a vector, matrix, or multidimensional array.
Data Types: single | double Complex Number Support: Yes
w — Weight0 (default) | 1 | vector
Weight, specified as one of:
0 — normalizes by the number of observations‐1. If there is only one observation, the weight is 1.
1 — normalizes by the number of observations.
a vector made up of nonnegative scalar weights corresponding to the dimension of A along which thevariance is calculated.
Data Types: single | double
dim — Dimension to operate alongpositive integer scalar
Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the defaultis the first array dimension whose size does not equal 1.
Dimension dim indicates the dimension whose length reduces to 1. The size(V,dim) is 1, while the sizesof all other dimensions remain the same.
Consider a twodimensional input array, A.
If dim = 1, then var(A,0,1) returns a row vector containing the variance of the elements in eachcolumn.
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If dim = 2, then var(A,0,2) returns a column vector containing the variance of the elements in eachrow.
var returns an array of zeros the same size as A when dim is greater than ndims(A).
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
nanflag — NaN condition‘includenan' (default) | ‘omitnan'
NaN condition, specified as one of these values:
‘includenan' — the variance of input containing NaN values is also NaN.
‘omitnan' — all NaN values appearing in either the input array or weight vector are ignored.
Data Types: char
More About
Variance
For a random variable vector A made up of N scalar observations, the variance is defined as
where μ is the mean of A,
Some definitions of variance use a normalization factor of N instead of N1, which can be specified bysetting w to 1. In either case, the mean is assumed to have the usual normalization factor N.
See Alsocorrcoef | cov | mean | std
Introduced before R2006a
V = 1N −1
N
i=1Ai − μ
2
μ = 1N
N
i=1Ai .