These are termed a weighted mean and standard deviation:
% % % References:
% % % https://en.wikipedia.org/wiki/Weighted_standard_deviation#Weighted_sample_variance
% % % http://itl.nist.gov/div898/software/dataplot/refman2/ch2/weightsd.pdf
%
% 10 (0.1) 11 (0.2) 12 (0.4) 13 (0.05) 14 (0.05) 15 (0.2)
X = 10:15; % Data
W = [0.1 0.2 0.4 0.05 0.05 0.2]; % Weights
N = length(X);
Wmean = X*W'/sum(W); % Weighted mean
Wstd = sqrt((sum(W.*(X - Wmean).^2)/N) / ((N-1)*sum(W)/N)); % Weighted std
