: Specifically, follows the chi-squared distribution with degrees of freedom, where is the number of dimensions of the normal distribution. If the number of dimensions is 2, for example, the probability of a particular calculated being less than some threshold is . To determine a threshold to achieve a particular probability, , use , for 2 dimensions. For number of dimensions other than 2, the cumulative chi-squared distribution should be consulted. In a normal distribution, the region where the Mahalanobis distance is less than one (i.e. the region inside the ellipsoid at distance one) is exactly the region where the probability distribution is concave. Mahalanobis distance is proportional, for a normal distribution, to the square root of the negative log-likelihood (after adding a constant so the minimum is at zero). Other forms of multivariate location and scatter The sample mean and covariance matrix can be quite sensitive to outliers, therefore other approaches to calculating the multivariate location and scatter of data are also commonly used when calculating the Mahalanobis distance. The Minimum Covariance Determinant approach estimates multivariate location and scatter from a subset numbering data points that has the smallest variance-covariance matrix determinant. The Minimum Volume Ellipsoid approach is similar to the Minimum Covariance Determinant approach in that it works with a subset of size data points, but the Minimum Volume Ellipsoid estimates multivariate location and scatter from the ellipsoid of minimal volume that encapsulates the data points. Each method varies in its definition of the distribution of the data, and therefore produces different Mahalanobis distances. The Minimum Covariance Determinant and Minimum Volume Ellipsoid approaches are more robust to samples that contain outliers, while the sample mean and covariance matrix tends to be more reliable with small and biased data sets. Relationship to normal random variables In general, given a normal (Gaussian) random variable with variance and mean , any other normal random variable (with mean and variance ) can be defined in terms of by the equation Conversely, to recover a normalized random variable from any normal random variable, one can typically solve for . If we square both sides, and take the square-root, we will get an equation for a metric that looks a lot like the Mahalanobis distance: The resulting magnitude is always non-negative and varies with the distance of the data from the mean, attributes that are convenient when trying to define a model for the