time   calls  line
                  1 function y = var(x,w,dim)
                  2 %VAR Variance.
                  3 %   For vectors, Y = VAR(X) returns the variance of the values in X.  For
                  4 %   matrices, Y is a row vector containing the variance of each column of
                  5 %   X.  For N-D arrays, VAR operates along the first non-singleton
                  6 %   dimension of X.
                  7 %
                  8 %   VAR normalizes Y by N-1 if N>1, where N is the sample size.  This is
                  9 %   an unbiased estimator of the variance of the population from which X is
                 10 %   drawn, as long as X consists of independent, identically distributed
                 11 %   samples. For N=1, Y is normalized by N. 
                 12 %
                 13 %   Y = VAR(X,1) normalizes by N and produces the second moment of the
                 14 %   sample about its mean.  VAR(X,0) is the same as VAR(X).
                 15 %
                 16 %   Y = VAR(X,W) computes the variance using the weight vector W.  W typically
                 17 %   contains either counts or inverse variances.  The length of W must equal
                 18 %   the length of the dimension over which VAR operates, and its elements must
                 19 %   be nonnegative.  If X(I) is assumed to have variance proportional to
                 20 %   1/W(I), then Y * MEAN(W)/W(I) is an estimate of the variance of X(I).  In
                 21 %   other words, Y * MEAN(W) is an estimate of variance for an observation
                 22 %   given weight 1.
                 23 %
                 24 %   Y = VAR(X,W,DIM) takes the variance along the dimension DIM of X.  Pass
                 25 %   in 0 for W to use the default normalization by N-1, or 1 to use N.
                 26 %
                 27 %   The variance is the square of the standard deviation (STD).
                 28 %
                 29 %   Example: If X = [4 -2 1
                 30 %                    9  5 7]
                 31 %      then var(X,0,1) is [12.5 24.5 18.0] and var(X,0,2) is [9.0
                 32 %                                                             4.0]
                 33 %
                 34 %   Class support for inputs X, W:
                 35 %      float: double, single
                 36 %
                 37 %   See also MEAN, STD, COV, CORRCOEF.
                 38 
                 39 %   VAR supports both common definitions of variance.  If X is a
                 40 %   vector, then
                 41 %
                 42 %      VAR(X,0) = SUM(RESID.*CONJ(RESID)) / (N-1)
                 43 %      VAR(X,1) = SUM(RESID.*CONJ(RESID)) / N
                 44 %
                 45 %   where RESID = X - MEAN(X) and N is LENGTH(X). For scalar X,
                 46 %   the first definition would result in NaN, so the denominator N 
                 47 %   is always used.
                 48 %
                 49 %   The weighted variance for a vector X is defined as
                 50 %
                 51 %      VAR(X,W) = SUM(W.*RESID.*CONJ(RESID)) / SUM(W)
                 52 %
                 53 %   where now RESID is computed using a weighted mean.
                 54 
                 55 %   Copyright 1984-2010 The MathWorks, Inc.
                 56 %   $Revision: 1.7.4.9 $  $Date: 2010/08/23 23:07:43 $
                 57 
          1673   58 if isinteger(x)  
                 59     error(message('MATLAB:var:integerClass'));
                 60 end
                 61 
  0.01    1673   62 if nargin < 2 || isempty(w), w = 0; end 
                 63 
          1673   64 if nargin < 3 
                 65     % The output size for [] is a special case when DIM is not given.
          1473   66     if isequal(x,[]), y = NaN(class(x)); return; end 
                 67 
                 68     % Figure out which dimension sum will work along.
          1473   69     dim = find(size(x) ~= 1, 1); 
          1473   70     if isempty(dim), dim = 1; end 
          1473   71 end 
          1673   72 n = size(x,dim); 
                 73 
                 74 % Unweighted variance
          1673   75 if isequal(w,0) || isequal(w,1) 
          1673   76     if w == 0 && n > 1 
                 77         % The unbiased estimator: divide by (n-1).  Can't do this
                 78         % when n == 0 or 1.
          1673   79         denom = n - 1; 
                 80     else
                 81         % The biased estimator: divide by n.
                 82         denom = n; % n==0 => return NaNs, n==1 => return zeros
                 83     end
                 84 
          1673   85     if n > 0 % avoid divide-by-zero 
  0.28    1673   86         xbar = sum(x, dim) ./ n; 
  0.87    1673   87         x = bsxfun(@minus, x, xbar); 
          1673   88     end 
  0.98    1673   89     y = sum(abs(x).^2, dim) ./ denom; % abs guarantees a real result 
                 90 
                 91     % Weighted variance
                 92 elseif isvector(w) && all(w >= 0)
                 93     if numel(w) ~= n
                 94         if isscalar(w)
                 95             error(message('MATLAB:var:invalidWgts'));
                 96         else
                 97             error(message('MATLAB:var:invalidSizeWgts'));
                 98         end
                 99     end
                100 
                101     % Normalize W, and embed it in the right number of dims.  Then
                102     % replicate it out along the non-working dims to match X's size.
                103     wresize = ones(1,max(ndims(x),dim)); wresize(dim) = n;
                104     w = reshape(w ./ sum(w), wresize);
                105     x0 = bsxfun(@times, w, x);
                106     x = bsxfun(@minus, x, sum(x0, dim));
                107     y = sum(bsxfun(@times, w, abs(x).^2), dim); % abs guarantees a real result
                108 
                109 else
                110     error(message('MATLAB:var:invalidWgts'));
                111 end