Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

rxy=0. 0 rxy=0. 1 rxy=0. 5

rxy=− 0. 7 rxy=− 0. 9 rxy=0. 99

x

y

Figure 31.1 Scatter plots for two-dimensional data samples of sizeN= 1000,

with various values of the correlationr. No scales are plotted, since the value

ofris unaffected by shifts of origin or changes of scale inxandy.

We may also define the closely relatedsample correlationby

rxy=

Vxy

sxsy

,

which can take values between−1 and +1. If thexiandyiare independent then

Vxy=0=rxy, and from (31.12) we see thatxy= ̄xy ̄. It should also be noted

that the value ofrxyis not altered by shifts in the origin or by changes in the

scale of thexioryi. In other words, ifx′=ax+bandy′=cy+d,wherea,

b,c,dare constants, thenrx′y′=rxy. Figure 31.1 shows scatter plots for several

two-dimensional random samplesxi,yiof sizeN= 1000, each with a different

value ofrxy.

Ten UK citizens are selected at random and their heights and weights are found to be as

follows (to the nearestcmorkgrespectively):

Person ABCDEFGH I J

Height (cm) 194 168 177 180 171 190 151 169 175 182

Weight (kg) 75 53 72 80 75 75 57 67 46 68

Calculate the sample correlation between the heights and weights.

In order to find the sample correlation, we begin by calculating the following sums (where
xiare the heights andyiare the weights)
∑

i

xi= 1757,

∑

i

yi= 668,