Editing Least squares (section)

==Statistical testing==
If the [[probability distribution]] of the parameters is known or an asymptotic approximation is made, [[confidence limits]] can be found. Similarly, statistical tests on the residuals can be conducted if the probability distribution of the residuals is known or assumed. We can derive the probability distribution of any linear combination of the dependent variables if the probability distribution of experimental errors is known or assumed. Inferring is easy when assuming that the errors follow a normal distribution, consequently implying that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.<ref name=":1" />

It is necessary to make assumptions about the nature of the experimental errors to test the results statistically. A common assumption is that the errors belong to a normal distribution. The [[central limit theorem]] supports the idea that this is a good approximation in many cases.

* The [[Gauss–Markov theorem]]. In a linear model in which the errors have [[expected value|expectation]] zero conditional on the independent variables, are [[uncorrelated]] and have equal [[variance]]s, the best linear [[unbiased]] estimator of any linear combination of the observations, is its least-squares estimator. "Best" means that the least squares estimators of the parameters have minimum variance. The assumption of equal variance is valid when the errors all belong to the same distribution.<ref>{{cite encyclopedia |last1=Hallin |first1=Marc |title=Gauss-Markov Theorem |url=https://onlinelibrary.wiley.com/doi/10.1002/9780470057339.vnn102 |publisher=Wiley |date=2012 |encyclopedia=Encyclopedia of Environmetrics |doi=10.1002/9780470057339.vnn102 |isbn=978-0-471-89997-6 |access-date=18 October 2023}}</ref>
*If the errors belong to a normal distribution, the least-squares estimators are also the [[maximum likelihood estimator]]s in a linear model.

However, suppose the errors are not normally distributed. In that case, a [[central limit theorem]] often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.  For this reason, given the important property that the error mean is independent of the independent variables, the distribution of the error term is not an important issue in regression analysis.  Specifically, it is not typically important whether the error term follows a normal distribution.