Mathematical Methods for Physics and Engineering - Matematica.NET

31.2 SAMPLE STATISTICSmoments of the sample. For example,n 3 = 1 N= 1 NN∑(x i − m 1 ) 3i=1N∑(x 3 i − 3m 1 x 2 i +3m 2 1x i − m 3 1)i=1= m 3 − 3m 1 m 2 +3m 2 1m 1 − m 3 1= m 3 − 3m 1 m 2 +2m 3 1, (31.11)which may be compared with equation (30.53) in the previous chapter.Mirroring our discussion of the normalised central moments γ r of a populationin subsection 30.5.5, we can also describe a sample in terms of the dimensionlessquantitiesg k = n k= n kn k/2 s k ;2g 3 and g 4 are called the sample skewness and kurtosis. Likewise, it is common todefine the excess kurtosis of a sample by g 4 − 3.31.2.4 Covariance and correlationSo far we have assumed that each data item of the sample consists of a singlenumber. Now let us suppose that each item of data consists of a pair of numbers,so that the sample is given by (x i ,y i ), i =1, 2,...,N.We may calculate the sample means, ¯x and ȳ, and sample variances, s 2 x ands 2 y,ofthex i and y i values individually but these statistics do not provide anymeasure of the relationship between the x i and y i . By analogy with our discussionin subsection 30.12.3 we measure any interdependence between the x i and y i interms of the sample covariance, which is given byV xy = 1 N∑(x i − ¯x)(y i − ȳ)Ni=1= (x − ¯x)(y − ȳ)= xy − ¯xȳ. (31.12)Writing out the last expression in full, we obtain the form most useful forcalculations, which reads(V xy = 1 N) (∑x i y i − 1 N)(∑ N)∑NN 2 x i y i .i=11227i=1i=1