Mahalanobis distance and Gaussian distribution

The Mahalanobis distance is a distance measure that accounts for the covariance or "stretch" of the shape in which the data lies.

$$D_{\text{Mahalanobis}}(\mathbf{x}, \mathbf{y}) = \sqrt{(\mathbf{x} - \mathbf{y})^T\Sigma^{-1}(\mathbf{x} - \mathbf{y})}$$

This is very similar to the exponential term in the Gaussian distribution.

It is useful to get an intuitive feel about the standard deviation. In the one-dimensional case, where
$$p(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma}}$$
the probability at one standard deviation away from the center is $\frac{1}{e}$ (37%) of the peak probability.

Source:
https://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg