Editing Studentized residual (section)

==Distribution==
{{distinguish-redirect|Tau distribution|Tau coefficient}}
If the errors are independent and  [[normal distribution|normally distributed]] with [[expected value]] 0 and variance ''σ''<sup>2</sup>, then the [[probability distribution]] of the ''i''th externally studentized residual <math>t_{i(i)}</math> is a [[Student's t-distribution]] with ''n''&nbsp;−&nbsp;''m''&nbsp;−&nbsp;1 [[degrees of freedom (statistics)|degrees of freedom]], and can range from <math>\scriptstyle-\infty</math> to <math>\scriptstyle+\infty</math>.

On the other hand, the internally studentized residuals are in the range <math> 0 \,\pm\, \sqrt{\nu}</math>, where ''ν'' = ''n''&nbsp;−&nbsp;''m'' is the number of residual degrees of freedom. If ''t''<sub>''i''</sub> represents the internally studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then:<ref name=NOAA>Allen J. Pope (1976), "The statistics of residuals and the detection of outliers", U.S. Dept. of Commerce, National Oceanic and Atmospheric Administration, National Ocean Survey, Geodetic Research and Development Laboratory, 136 pages, [http://www.ngs.noaa.gov/PUBS_LIB/TRNOS65NGS1.pdf], eq.(6)</ref>

:<math>t_i \sim \sqrt{\nu} {t \over \sqrt{t^2+\nu-1}}</math>

where ''t'' is a random variable distributed as [[Student's t-distribution]] with ''ν''&nbsp;−&nbsp;1 degrees of freedom. In fact, this implies that ''t''<sub>''i''</sub><sup>2</sup> /''ν'' follows the [[beta distribution]] ''B''(1/2,(''ν''&nbsp;−&nbsp;1)/2).
The distribution above is sometimes referred to as the '''tau distribution''';<ref name=NOAA/> it was first derived by Thompson in 1935.<ref name=Thompson>{{cite journal|last1=Thompson|first1=William R.|title=On a Criterion for the Rejection of Observations and the Distribution of the Ratio of Deviation to Sample Standard Deviation|journal=The Annals of Mathematical Statistics|date=1935|volume=6|issue=4|pages=214–219|doi=10.1214/aoms/1177732567|doi-access=free}}</ref>

When ''ν'' = 3, the internally studentized residuals are [[uniform distribution (continuous)|uniformly distributed]] between <math>\scriptstyle-\sqrt{3}</math> and <math>\scriptstyle+\sqrt{3}</math>.
If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't apply. In this case, the ''t''<sub>''i''</sub> are all either +1 or −1, with 50% chance for each.

The standard deviation of the distribution of internally studentized residuals is always 1, but this does not imply that the standard deviation of all the ''t''<sub>''i''</sub> of a particular experiment is 1.
For instance, the internally studentized residuals when fitting a straight line going through (0, 0) to the points (1, 4), (2, −1), (2, −1) are <math>\sqrt{2},\ -\sqrt{5}/5,\ -\sqrt{5}/5</math>, and the standard deviation of these is not 1.

Note that any pair of studentized residual ''t''<sub>''i''</sub> and ''t''<sub>''j''</sub> (where {{nowrap|<math>i \neq j</math>),}} are NOT i.i.d. They have the same distribution, but are not independent due to constraints on the residuals having to sum to 0 and to have them be orthogonal to the design matrix.