Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICS◮Calculate ∑ Ni=1 x i and ∑ Ni=1 x2 i for the data given in table 31.1 and hence find the meanand standard deviation of the sample.From table 31.1, we obtainN∑x i = 188.7 + 204.7+···+ 200.0 = 1479.8,i=1N∑x 2 i = (188.7) 2 + (204.7) 2 + ···+ (200.0) 2 = 275 334.36.i=1Since N = 8, we find as before (quoting the final results to one decimal place)¯x = 1479.88√( ) 2275 334.36 1479.8= 185.0, s =−=14.2. ◭8831.2.3 Moments and central momentsBy analogy with our discussion of probability distributions in section 30.5, thesample mean and variance may also be described respectively as the first momentand second central moment of the sample. In general, for a sample x i , i =1, 2,...,N, we define the rth moment m r and rth central moment n r asm r = 1 Nn r = 1 NN∑x r i , (31.9)i=1N∑(x i − m 1 ) r . (31.10)i=1Thus the sample mean ¯x and variance s 2 may also be written as m 1 and n 2respectively. As is common practice, we have introduced a notation in whicha sample statistic is denoted by the Roman letter corresponding to whicheverGreek letter is used to describe the corresponding population statistic. Thus, weuse m r and n r to denote the rth moment and central moment of a sample, sincein section 30.5 we denoted the rth moment and central moment of a populationby µ r and ν r respectively.This notation is particularly useful, since the rth central moment of a sample,m r , may be expressed in terms of the rth- and lower-order sample moments n r in away exactly analogous to that derived in subsection 30.5.5 for the correspondingpopulation statistics. As discussed in the previous section, the sample variance isgiven by s 2 = x 2 − ¯x 2 but this may also be written as n 2 = m 2 − m 2 1 ,whichistobecompared with the corresponding relation ν 2 = µ 2 −µ 2 1 derived in subsection 30.5.3for population statistics. This correspondence also holds for higher-order central1226