(Some) mathematical measures

This page outlines, mainly for convenience, some useful measures/metrics utilised for several purposes.

The harmonic mean

It is the reciprocal of the mean of reciprocals:

h=ni=1i=n1xih = \frac{n}{\sum_{i=1}^{i=n} \frac{1}{x_i}}

Common measures of distance/similarity


The euclidean distance of two vectorsva=(xi)av_a = (x^i)_aandvb=(xi)bv_b = (x^i)_bis the norml2l_2of the vector connecting them (it measures its length):

d=i(xaixbi)2=vAvB2d = \sqrt{\sum_i (x^i_a - x^i_b)^2} = ||v_A - v_B||_2


The Hamming distance expresses the number of different elements in two lists/strings:

A=110101;B=111001;dAB=2A = 110101; B = 111001; d_{AB} = 2

Jaccard (index)

Given two finite sets A and B, the Jaccard index gives a measure of how much they overlap, as

JAB=ABABJ_{AB} = \frac{|A \cap B|}{|A \cup B|}


Also called cityblock, the Manhattan distance between two points is the norml1l_1of the shortest path a car would take between these two points in Manhattan (which has a grid layout):

d=iuivid = \sum_i |u_i - v_i|


The Minkowski distance is a generalisation of both the euclidean and the Manhattan to a generic p:

d=(ixiyip)1/pd = \left(\sum_i |x_i - y_i|^p\right)^{1/p}


The cosine similarity is given by the cosine of the angleθ\thetaspanned by the two vectors

d=cosθ=uˉvˉuˉvˉd = \cos \theta = \frac{\bar u \cdot \bar v}{|\bar u| |\bar v|}

So two perfectly overlapping vectors would have a cosine similarity of 1 and vectors at 9090^{\circ}would have a cosine similarity of 0.


It is also called chessboard distance. In the game of chess, the Chebyshev distance between the centers of the squares is the minimum number of moves a king needs to go from a square to another one.

maxiuivi\max_i |u_i - v_i|

See the figure here, it reports in red all the Chebyshev distance value from where the king (well, there's a drawing for it ...) sits to cell; note that the king can move horizontally, vertically and diagonally.