Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

Calculate

∑N

i=1xiand

∑N

i=1x

2

ifor the data given in table 31.1 and hence find the mean

and standard deviation of the sample.

From table 31.1, we obtain

∑N

i=1

xi= 188.7 + 204.7+···+ 200.0 = 1479. 8 ,

∑N

i=1

x^2 i= (188.7)^2 + (204.7)^2 +···+ (200.0)^2 = 275 334. 36.

SinceN= 8, we find as before (quoting the final results to one decimal place)

̄x=

1479. 8

8

= 185. 0 ,s=

√

275 334. 36

8

−

(

1479. 8

8

) 2

=14. 2 .

31.2.3 Moments and central moments

By analogy with our discussion of probability distributions in section 30.5, the

sample mean and variance may also be described respectively as the first moment

and second central moment of the sample. In general, for a samplexi,i=

1 , 2 ,...,N, we define therth momentmrandrth central momentnras

mr=

1

N

∑N

i=1

xri, (31.9)

nr=

1

N

∑N

i=1

(xi−m 1 )r. (31.10)

Thus the sample meanx ̄and variances^2 may also be written asm 1 andn 2

respectively. As is common practice, we have introduced a notation in which

a sample statistic is denoted by the Roman letter corresponding to whichever

Greek letter is used to describe the corresponding population statistic. Thus, we

usemrandnrto denote therth moment and central moment of a sample, since

in section 30.5 we denoted therth moment and central moment of a population

byμrandνrrespectively.

This notation is particularly useful, since therth central moment of a sample,

mr, may be expressed in terms of therth- and lower-order sample momentsnrin a

way exactly analogous to that derived in subsection 30.5.5 for the corresponding

population statistics. As discussed in the previous section, the sample variance is

given bys^2 =x^2 − ̄x^2 but this may also be written asn 2 =m 2 −m^21 ,whichistobe

compared with the corresponding relationν 2 =μ 2 −μ^21 derived in subsection 30.5.3

for population statistics. This correspondence also holds for higher-order central