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A Gaussian error is a random error term with a Normal distribution. The exact least-squares conclusions require a specified model: for fixed-design linear regression, letwhere the $n\times p$ design matrix $X$ has full column rank. The error vector thus has independent zero-mean Gaussian coordinates of common variance. For fixed ${\sigma }^2$, maximizing the likelihood over $\beta$ is equivalent to minimizing $\Vert Y-X\beta {\Vert }^2$.

The least-squares estimator and its exact sampling law areThis follows because a linear transformation of a Gaussian vector is Gaussian. It is an exact finite-sample law, rather than merely Asymptotic normality.

Zero mean and common variance alone do not imply Gaussian errors or exact normal coefficient estimates. Even with Gaussian errors, fitted residuals are not independent copies of them: with $H=X(X^{T}X{)}^{-1}{X}^T$, the residual vector is $(I-H)\epsilon$ and has covariance ${\sigma }^2(I-H)$, usually with correlated coordinates and reduced rank. A Normality assumption must identify whether it concerns errors, responses or another latent variable.

Use the discussion of normality assumption in part b of the Experimental Psychology paper. The relevant distribution depends on the statistical model and procedure.

Use a Student t-test to assess a mean or mean difference relative to its standard error. A one-sample version compares a population mean with a specified reference. An independent two-sample version compares two separate groups, such as groups assigned different instructions; Welch's t-test permits unequal group variances. A paired t-test uses within-pair differences, such as each participant's before/after scores, rather than pretending those measurements are independent.^[ref]

The relevant normality assumption concerns within-group errors for the independent-groups model or the distribution of within-pair differences for the paired model. Independent participants or pairs remain essential. Report the estimated difference and its confidence interval together with the test result. A small p-value indicates incompatibility with the specified null hypothesis under the model; it does not measure the size of the difference, the probability that the null hypothesis is true, or whether the difference matters practically.

The normality assumption concerns the model's error distribution, not necessarily the pooled raw observations. In an independent-groups analysis of variance, errors around each group mean are modeled as following a normal distribution; combining groups with genuinely different means can produce a nonnormal pooled distribution even when this assumption holds. In a paired t-test, the relevant distribution is that of within-pair differences. Linear regression conventionally assumes normal errors conditional on the predictors for exact small-sample Student t-tests and F-tests.^[ref]

Inspect suitable within-group or fitted-model regression residuals using a quantile-quantile plot and examine outliers and asymmetry. A formal normality test is not a decisive automatic gatekeeper: with small samples it may miss important departures, while large samples can detect minor departures. Approximate robustness depends on the sample size, balance, tail behavior and intended inference. The central limit theorem concerns sampling distributions and does not make every raw observation normal or repair dependence. Depending on the question, a justified transformation, a better-fitting probability model, robust inference or an appropriate bootstrap may be preferable. The normality assumption, independence and equal variance are distinct assumptions.