APPENDIX A1

Summation Notation

A1.1 SUMMATION AND ITS PROPERTIES

The addition of numbers is basic to our study of statistics. To avoid a detailed and repeated writing of this operation, the symbol Σ (the Greek capital letter sigma) is used as mathematical shorthand for the operation of addition.

Summation Notation Σ

The notation images represents the sum of n numbers x1, x2, …, xn and is read as the sum of all xi with i ranging from 1 to n.

images

The term following the sign Σ indicates the quantities that are being summed, and the notations on the bottom and the top of the Σ specify the range of the terms being added. For instance,

images

Example

Suppose that the four measurements in a data set are given as x1 = 2, x2 = 5, x3 = 3, x4 = 4. Compute the numerical values of

images

SOLUTION

images

Alternatively, noting that (xi − 3)2 = images − 6xi + 9, we can write

A few basic properties of the summation operation are apparent ...

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