Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.3 ESTIMATORS AND SAMPLING DISTRIBUTIONS

∑

i

x^2 i= 310 041,

∑

i

y^2 i= 45 746,

∑

i

xiyi= 118 029.

ThesampleconsistsofN= 10 pairs of numbers, so the means of thexiand of theyiare
given by ̄x= 175.7and ̄y=66.8. Also,xy= 11 802.9. Similarly, the standard deviations
of thexiandyiare calculated, using (31.8), as

sx=

√

310 041

10

−

(

1757

10

) 2

=11. 6 ,

sy=

√

45 746

10

−

(

668

10

) 2

=10. 6.

Thus the sample correlation is given by

rxy=

xy−x ̄ ̄y

sxsy

=

11 802. 9 −(175.7)(66.8)

(11.6)(10.6)

=0. 54.

Thus there is a moderate positive correlation between the heights and weights of the
people measured.

It is straightforward to generalise the above discussion to data samples of

arbitrary dimension, the only complication being one of notation. We choose

to denote theith data item from ann-dimensional sample as (x(1)i,x(2)i ,...,x(in)),

where the bracketted superscript runs from 1 tonand labels the elements within

a given data item whereas the subscriptiruns from 1 toNand labels the data

items within the sample. In thisn-dimensional case, we can define thesample

covariance matrixwhose elements are

Vkl=x(k)x(l)−x(k)x(l)

and thesample correlation matrixwith elements

rkl=

Vkl

sksl

.

Both these matrices are clearly symmetric but arenotnecessarily positive definite.

31.3 Estimators and sampling distributions

In general, the populationP(x) from which a samplex 1 ,x 2 ,...,xNis drawn

isunknown.Thecentral aimof statistics is to use the sample valuesxito infer

certain properties of the unknown populationP(x), such as its mean, variance and

higher moments. To keep our discussion in general terms, let us denote the various

parameters of the population bya 1 ,a 2 ,..., or collectively bya. Moreover, we make

the dependence of the population on the values of these quantities explicit by

writing the population asP(x|a). For the moment, we are assuming that the

sample valuesxiare independent and drawn from the same (one-dimensional)

populationP(x|a), in which case

P(x|a)=P(x 1 |a)P(x 2 |a)···P(xN|a).