Statistics: Principles and Methods, 7th Edition

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 represents the sum of n numbers x₁, x₂, …, x_n and is read as the sum of all x_i with i ranging from 1 to n.

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,

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

SOLUTION

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

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