Mahalanobis Distance Calculator

How it is calculated

For a two-variable point x = (x1, x2), mean m = (m1, m2) and covariance matrix S = [[s11, s12], [s12, s22]], the tool computes D² = dᵀ S⁻¹ d with d = x − m, then D = √D². The 2×2 inverse is built from the determinant det = s11·s22 − s12²: S⁻¹ = (1/det) · [[s22, −s12], [−s12, s11]]. s11 and s22 are the variances of each variable, s12 is their covariance, all in the squared units of the data. Both D and D² are shown with four decimals. The covariance matrix must be positive definite: s11 and s22 above zero and a determinant above zero, that is s12² < s11·s22. Otherwise (including perfectly correlated variables) no result is shown. All seven fields are required.

Common mistakes

• Entering standard deviations in s11 and s22. These fields take variances (standard deviation squared); a standard deviation of 2 means s11 = 4.
• Entering the correlation coefficient (between −1 and 1) as s12. The covariance is the correlation multiplied by both standard deviations.
• Using an impossible covariance matrix. It needs s12² < s11·s22; with s11 = 4 and s22 = 9, any s12 of 6 or more in size (determinant zero or negative) gives no result.
• Dropping the sign of the covariance. With the same point and mean, s12 = 3 gives D = 1.1547 but s12 = −3 gives D = 2.0000, because the direction of the correlation changes which points count as typical.
• Reading D as D². The tool shows both, and a chi-square cut-off for two variables applies to D², not D.