Editing Errors and residuals (section)

===Remark===
It is remarkable that the [[Squared deviations|sum of squares of the residuals]] and the sample mean can be shown to be independent of each other, using, e.g. [[Basu's theorem]].<!-- Basu's theorem is definitely overkill in this case. It can be proved by far simpler methods. --> That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic:

:<math> T = \frac{\overline{X}_n - \mu_0}{S_n/\sqrt{n}}, </math>

where <math>\overline{X}_n - \mu_0</math> represents the errors, <math>S_n</math> represents the sample standard deviation for a sample of size ''n'', and unknown ''σ'', and the denominator term <math>S_n/\sqrt n</math> accounts for the standard deviation of the errors according to:<ref name="modernintro">{{Cite book|title=A modern introduction to probability and statistics : understanding why and how|date=2005-06-15|publisher=Springer London|author1=Frederik Michel Dekking|author2=Cornelis Kraaikamp|author3=Hendrik Paul Lopuhaä|author4=Ludolf Erwin Meester|isbn=978-1-85233-896-1|location=London|oclc=262680588}}</ref>

<math display="block">\operatorname{Var}\left(\overline{X}_n\right) = \frac{\sigma^2} n</math>

The [[probability distribution]]s of the numerator and the denominator separately depend on the value of the unobservable population standard deviation ''σ'', but ''σ'' appears in both the numerator and the denominator and cancels. That is fortunate because it means that even though we do not know&nbsp;''σ'', we know the probability distribution of this quotient: it has a [[Student's t-distribution]] with ''n''&nbsp;−&nbsp;1 degrees of freedom. We can therefore use this quotient to find a [[confidence interval]] for&nbsp;''μ''. This t-statistic can be interpreted as "the number of standard errors away from the regression line."<ref>{{Cite book|title=Practical statistics for data scientists : 50 essential concepts|author1=Peter Bruce|author2=Andrew Bruce|isbn=978-1-4919-5296-2|edition=First|publisher=O'Reilly Media Inc|location=Sebastopol, CA|oclc=987251007|date=2017-05-10}}</ref>