618 Chapter 21Probability and statistics
21.9 More than one variable
In many experiments in the physical sciences, the outcomes consist of the values
of more than one quantity at a time. For example, a kinetics experiment involves
the measurement of pairs (c, t) of concentration and time, in thermodynamics we
may have the pairs (p, T), in dynamics we may be following the flight of a body by
measuring (x, y, z, t) for position and time. Alternatively, the four values may be
height, IQ, mark in a maths examination, and personal best time (PB) in the 100 m
breaststroke for each member of a class of chemistry students.
Covariance and correlation
We consider an experiment in which each outcome consists of values of a pair of
variables(x, y): (x
1, y
1), (x
2, y
2), =, (x
N, y
N)for a sample of Npairs. We can compute
the means,EandF, variances,V(x)andV(y), and standard deviations,σ
xandσ
y,
of the separate variables. We can also consider the two variables together and
determine whether they are dependent on one another. A measure of the dependence
of two variables is the covariance, defined by
(21.46)
where. If the variables are independentthen 1 =1EF(each value of x
can occur with each value of y) and the covariance is zero. If values of xgreater than
the mean tend to occur with above average values of y, the covariance is positive
because the terms(x
i1 −1E)(y
i1 −1F)are positive. If large values of xoccur with small
values of y, the covariance is negative.
An alternative measure of dependence is the correlation coefficient
(21.47)
This quantity is dimensionless, with values − 11 ≤ 1 ρ
x,y1 ≤ 1 + 1. The variables xand y
are uncorrelatedwhen ρ
x,y1 = 10 , show positive correlationifρ
x,yis positive, and
negative correlationwhen it is negative. Ifρ
x,y1 = 1 ± 1 , xand yare completely (100%)
correlated, and a functional relation may exist between the variables. The scatter plots
in Figure 21.9 show some examples of correlation.
ρ
σσ σσ
xyxy xyxy xy xy
,=
,
=
cov( ) −
xy
xyN
xy
ii=
∑
1
cov( )xy ( )( )
N
xxyy xyxy
iii,= − −= −
∑
1
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
(a)ρ
x,y=0. 03
..........................................................................................................................................................................................................................................................................................................................................................
..
..
...
...
..
...
..
..
...
...
..
...
..
..
...
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
......................................................................................................................................................................... .......
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
.
(b)ρ
x,y=0. 52
.........................................................................................................................................................................................................................................................................................................................................................
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
......................................................................................................................................................................... .....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(c)ρ
x,y=0. 76
..........................................................................................................................................................................................................................................................................................................................................................
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
..
...
.......................................................................................................................................................................... .....
.
.
.
.
.
..
.
..
.
.
.
....
..
.
..
.
.
.
.
.
.
.
..
.
.
.
..
..
.
.
.
..
.
.
.
.
..
..
.
.
.
.
.
.
(d)ρ
x,y=0. 99
Figure 21.9