A Level Maths Notes: S3 – Spearman's Rank Correlation Coeffcient
The correlation coefficient
measurements the goodness of fit of a set of points
to
a straight line but two quantities may in fact observe a perfect
correlation without all the points lying on a straight line. The
quantities x and y have a perfect though non linear relationship as
shown on the graph below. The relationship is perfect for the ranges
of x and y in that if x increases then so does y.
The correlation between
variables
and
that
have a non linear relationship may be better expressed by a function
of the variables called the rank correlation coefficient. The rank
correlation coefficient is found by ordering the
in
terms of size, labelling them
Each
has
a corresponding
but
when the
are
put into order the
may
not have the same order as the
Suppose
thehave
ranks
Label
the ranks of theas
(though
we may in fact take them as
).
For each
find
and
find
in
some order and each of these is equal to
so
in
some order and each of these is equal to
and similarly for
Spearman's Rank correlation
coefficient is given by