Mahalanobis distance computes distance of two points considering covariance of data points, namely,

mahalanobis distance = (d – AVG(d)) / Covariance = d’C-1d where d is euclidean distance between two points.

In R[1]:

> x = data.frame(c(1, 0, 8, 2, 5), c(12, 14, 0, 2, 1), c(5, 7, 6, 3, 8)) 
> names(x) = c(“x”, “y”, “z”)
> x
  x  y z
1 1 12 5
2 0 14 7
3 8  0 6
4 2  2 3
5 5  1 8

> s = cov(x)
> s
       x      y    z
x  10.70 -17.95 1.55
y -17.95  44.20 0.95
z   1.55   0.95 3.70

Now we compute mahalanobis distance between the first data and the rest.

> mahalanobis(x, c(1, 12, 5), s)
[1] 0.000000 1.750912 4.585126 5.010909 7.552592

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References)
[1] Data is from 박광배, “다변량분석”, 학지사, p57.