Abstract: With the case of the quadratic polynomial outputs withoutcross-term, the correlated and uncorrelated contributions by correlatedinput variables to variance of output response are derived analyticallythrough the properties of the multi-dimensional correlated normaldistribution and conditional distribution. The results of examplesdemonstrate that the derived analytical expressions are correct. The derivedanalytical expressions can be used directly in recognition of thecontribution by input variables in quadratic or one-order polynomial outputwithout cross terms and can be compared for other new algorithms. Thismethod can also be extended to higher order polynomial with cross terms, tosolve the recognition of contribution by input variables in more complicatedoutputs.