Least absolute deviations
![](/Images/godic/202501/23/Linear_regression.svg0623.png")
![Figure A: A set of data points with reflection symmetry and multiple least absolute deviations solutions. The “solution area” is shown in green. The vertical blue lines represent the absolute errors from the pink line to each data point. The pink line is one of infinitely many solutions within the green area.](/Images/godic/202501/23/Least_absolute_deviations_regression_method_diagram0624.gif")
Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute value (LAV), least absolute residual (LAR), sum of absolute deviations, or the L1 norm condition, is a statistical optimality criterion and the statistical optimization technique that relies on it. Similar to the popular least squares technique, it attempts to find a function which closely approximates a set of data. In the simple case of a set of (x,y) data, the approximation function is a simple "trend line" in two-dimensional Cartesian coordinates. The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical "residuals" between points generated by the function and corresponding points in the data). The least absolute deviations estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution.