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A Gaussian error is an error term with a normal distribution. Independent Gaussian errors of equal variance turn least squares into maximum likelihood and make standardized linear coefficient estimators exactly normal. Zero mean and common variance alone do not imply Gaussian errors.

The least squares approach chooses parameters by minimizing the sum of squared residuals. A linear least-squares problem has affine residuals, whereas nonlinear least squares permits nonlinear dependence on the parameters. The Euclidean norm makes the objective $\frac{1}{2}\Vert R(x){\Vert }_2^2$; the factor one-half simplifies its derivatives without changing its minimizers.

Minimizing the squared residual norm gives the normal equationsFull column rank makes $X^{T}X$ invertible. With columns corresponding to $1,x,x^2-4$, the design matrix isSymmetry gives zero cross products between the odd column and either even column, and $\sum (x^2-4)=28-28=0$ gives the remaining orthogonality. Their squared norms are seven, 28 and 84, so $X^{T}X=\mathrm{diag}(7,28,84)$. ThereforeSince the error covariance is ${\sigma }^{2}I$, least squares has covariance ${\sigma }^2(X^{T}X{)}^{-1}$. ThusThe estimators have zero covariances. independence would require an additional distributional assumption such as Gaussian errors; the stated mean and variance assumptions alone give only the covariance calculation.