# Simple linear regression

**simple regressioni.e. regression linelinear least squares regression with an intercept term and a single explanatorstandard error of the slope coefficient**

In statistics, simple linear regression is a linear regression model with a single explanatory variable.wikipedia

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### Linear regression

**regression coefficientmultiple linear regressionregression**

In statistics, simple linear regression is a linear regression model with a single explanatory variable.

The case of one explanatory variable is called simple linear regression.

### Ordinary least squares

**OLSleast squaresOrdinary least squares regression**

It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible.

The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation.

### Theil–Sen estimator

**Kendall slopeMedian slopeRobust simple linear regression**

Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil–Sen estimator (which chooses a line whose slope is the median of the slopes determined by pairs of sample points).

In non-parametric statistics, the Theil–Sen estimator is a method for robustly fitting a line to sample points in the plane (simple linear regression) by choosing the median of the slopes of all lines through pairs of points.

### Deming regression

**Demingline of best orthogonal fitOrthogonal distance regression**

Deming regression (total least squares) also finds a line that fits a set of two-dimensional sample points, but (unlike ordinary least squares, least absolute deviations, and median slope regression) it is not really an instance of simple linear regression, because it does not separate the coordinates into one dependent and one independent variable and could potentially return a vertical line as its fit.

It differs from the simple linear regression in that it accounts for errors in observations on both the x- and the y- axis.

### Coefficient of determination

**R-squaredR'' 2 R 2**

The coefficient of determination ("R squared") is equal to r_{xy}^2 when the model is linear with a single independent variable.

One class of such cases includes that of simple linear regression where r 2 is used instead of R 2.

### Homoscedasticity

**homoscedastichomogeneity of variancehomoskedastic**

It is also possible to evaluate the properties under other assumptions, such as inhomogeneity, but this is discussed elsewhere.

As used in describing simple linear regression analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss–Markov theorem) is that the standard deviations of the error terms are constant and do not depend on the x-value.

### Line fitting

**line of best fitfitting a line**

### Linear trend estimation

**trendTrend estimationtrends**

This can always be done in closed form since this is a case of simple linear regression.

### Pearson correlation coefficient

**correlation coefficientPearson product-moment correlation coefficientPearson correlation**

In this case, the slope of the fitted line is equal to the correlation between y and x corrected by the ratio of standard deviations of these variables.

In this case, it estimates the fraction of the variance in Y that is explained by X in a simple linear regression.

### Segmented regression

**Linear segmented regressionPiecewise regressionsegmented regression analysis**

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, simple linear regression is a linear regression model with a single explanatory variable.

### Dependent and independent variables

**dependent variableindependent variableexplanatory variable**

That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables.

### Cartesian coordinate system

**Cartesian coordinatesCartesian coordinateCartesian**

That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables.

### Line (geometry)

**linestraight linelines**

That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables.

### Errors and residuals

**residualserror termresidual**

It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible.

### Least absolute deviations

**Least absolute deviationLeast absolute errorsLAD**

Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil–Sen estimator (which chooses a line whose slope is the median of the slopes determined by pairs of sample points).

### Slope

**gradientslopesgradients**

Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil–Sen estimator (which chooses a line whose slope is the median of the slopes determined by pairs of sample points).

### Median

**averagesample medianmedian-unbiased estimator**

### Mathematical model

**mathematical modelingmodelmathematical models**

Consider the model function

### Standard score

**normalizednormalisedz-score**

is the slope of the regression line of the standardized data points (and that this line passes through the origin).

### Correlation and dependence

**correlationcorrelatedcorrelations**

See sample correlation coefficient for additional details.

### Statistical model

**modelprobabilistic modelstatistical modeling**

Description of the statistical properties of estimators from the simple linear regression estimates requires the use of a statistical model.

### Bias of an estimator

**unbiasedunbiased estimatorbias**

The estimators and are unbiased.

### Confidence interval

**confidence intervalsconfidence levelconfidence**

Confidence intervals were devised to give a plausible set of values to the estimates one might have if one repeated the experiment a very large number of times.

### Central limit theorem

**Lyapunov's central limit theoremlimit theoremscentral limit**

The latter case is justified by the central limit theorem.