# Studentized residual

**studentized residualsexternallystudentization of residualsTau-distribution**

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation.wikipedia

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### Errors and residuals

**residualserror termresidual**

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing.

One can standardize statistical errors (especially of a normal distribution) in a z-score (or "standard score"), and standardize residuals in a t-statistic, or more generally studentized residuals.

### William Sealy Gosset

**StudentGossetGosset, William Sealy**

It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student.

Although introduced by others, Studentized residuals are named in Student's honour because, like the problem that led to Student's t-distribution, the idea of adjusting for estimated standard deviations is central to that concept.

### T-statistic

**Student's t-statistict''-statisticStudent's ''t''-statistic**

It is a form of a Student's t-statistic, with the estimate of error varying between points.

### Normalization (statistics)

**normalizednormalisedaverage**

Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

### Outlier

**outliersstatistical outliersconservative estimate**

This is an important technique in the detection of outliers.

### Leverage (statistics)

**leveragehigh-leverage pointleverage score**

The leverage h ii is the ith diagonal entry in the hat matrix.

The corresponding studentized residual—the residual adjusted for its observation-specific estimated residual variance—is then

### Projection matrix

**hat matrixannihilator matrixobservation matrix**

and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:

### Student's t-distribution

**Student's ''t''-distributiont-distributiont''-distribution**

If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation.

### Unit of observation

**data pointdata pointsobservation**

Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing.

### Regression analysis

**regressionmultiple regressionregression model**

Typically the standard deviations of residuals in a sample vary greatly from one data point to another even when the errors all have the same standard deviation, particularly in regression analysis; thus it does not make sense to compare residuals at different data points without first studentizing. The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal.

### Student

**studentscollege studentpupils**

It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym Student.

### Standard deviation

**standard deviationssample standard deviationSD**

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

### Standard score

**normalizednormalisedz-score**

Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing.

### Joint probability distribution

**joint distributionjoint probabilitymultivariate distribution**

The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal.

### Independence (probability theory)

**independentstatistically independentindependence**

where the errors, are independent and all have the same variance \sigma^2.

### Robust statistics

**robustbreakdown pointrobustness**

It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence.

### Linear regression

**regression coefficientmultiple linear regressionregression**

It is also reflected in the influence functions of various data points on the regression coefficients: endpoints have more influence. Consider the simple linear regression model

### Estimator

**estimatorsestimateestimates**

In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

### Univariate distribution

**univariateuni**

It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.

### Design matrix

**data matrixdesign matricesdata matrices**

For this simple model, the design matrix is

### Projection (linear algebra)

**orthogonal projectionprojectionprojection operator**

and the hat matrix H is the matrix of the orthogonal projection onto the column space of the design matrix:

### Normal distribution

**normally distributedGaussian distributionnormal**

If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

### Expected value

**expectationexpectedmean**

If the errors are independent and normally distributed with expected value 0 and variance σ 2, then the probability distribution of the ith externally studentized residual t_{i(i)} is a Student's t-distribution with n − m − 1 degrees of freedom, and can range from to.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**