Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.2 SAMPLE STATISTICS

moments of the sample. For example,

n 3 =

1

N

∑N

i=1

(xi−m 1 )^3

=

1

N

∑N

i=1

(x^3 i− 3 m 1 x^2 i+3m^21 xi−m^31 )

=m 3 − 3 m 1 m 2 +3m^21 m 1 −m^31

=m 3 − 3 m 1 m 2 +2m^31 , (31.11)

which may be compared with equation (30.53) in the previous chapter.

Mirroring our discussion of the normalised central momentsγrof a population

in subsection 30.5.5, we can also describe a sample in terms of the dimensionless

quantities

gk=

nk

nk/ 22

=

nk

sk

;

g 3 andg 4 are called the sample skewness and kurtosis. Likewise, it is common to

define theexcesskurtosis of a sample byg 4 −3.

31.2.4 Covariance and correlation

So far we have assumed that each data item of the sample consists of a single

number. Now let us suppose that each item of data consists of a pair of numbers,

so that the sample is given by (xi,yi),i=1, 2 ,...,N.

We may calculate the sample means,x ̄and ̄y, and sample variances,s^2 xand

s^2 y,ofthexiandyivalues individually but these statistics do not provide any

measure of the relationship between thexiandyi. By analogy with our discussion

in subsection 30.12.3 we measure any interdependence between thexiandyiin

terms of thesample covariance, which is given by

Vxy=

1

N

∑N

i=1

(xi− ̄x)(yi− ̄y)

=(x− ̄x)(y−y ̄)

=xy− ̄x ̄y. (31.12)

Writing out the last expression in full, we obtain the form most useful for

calculations, which reads

Vxy=

1

N

(N

∑

i=1

xiyi

)

−

1

N^2

(N

∑

i=1

xi

)(N

∑

i=1

yi

)

.