A computationally efficient way to compute the sample mean

Chapter , Problem 18

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A computationally efficient way to compute the sample mean and sample variance of the data set x1, x2, . . . , xn is as follows. Let x j = j _ i=1 xi j , j =1, . . . , n be the sample mean of the first j data values, and let s2 j = j _ i=1 (xi xj)2 j 1 , j =2, . . . , n be the sample variance of the first j, j 2, values. Then, with s2 1 =0, it can be shown that xj+1 = xj + xj+1 xj j +1 and s2 j+1 = 1 1 j s2 j +(j +1)(xj+1 xj)2 (a) Use the preceding formulas to compute the sample mean and sample variance of the data values 3, 4, 7, 2, 9, 6. (b) Verify your results in part (a) by computing as usual. (c) Verify the formula given above for xj+1 in terms of x j .