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1.11: Summation Notation

  • Page ID
  • Learning Objectives

    • Use summation notation to express the sum of all numbers
    • Use summation notation to express the sum of a subset of numbers
    • Use summation notation to express the sum of squares

    Many statistical formulas involve summing numbers. Fortunately there is a convenient notation for expressing summation. This section covers the basics of this summation notation.

    Let's say we have a variable \(X\) that represents the weights (in grams) of \(4\) grapes. The data are shown in Table \(\PageIndex{1}\).

    Table \(\PageIndex{1}\): Weights of \(4\) grapes.
    Grape X
    1 4.6
    2 5.1
    3 4.9
    4 4.4

    We label Grape \(1's\) weight \(X_1\), Grape \(2's\) weight \(X_2\), etc. The following formula means to sum up the weights of the four grapes:

    \[ \sum_{i=1}^4 X_i \]

    The Greek letter capital sigma (\(\sum\)) indicates summation. The "\(i = 1\)" at the bottom indicates that the summation is to start with \(X_1\) and the \(4\) at the top indicates that the summation will end with \(X_4\). The "\(X_i\)" indicates that \(X\) is the variable to be summed as \(i\) goes from \(1\) to \(4\). Therefore,

    \[ \sum_{i=1}^4 X_i = X_1 + X_2 + X_3 + X_4 = 4.6 + 5.1 + 4.9 + 4.4 = 19.0 \]

    The symbol

    \[ \sum_{i=1}^3 X_i \]

    indicates that only the first \(3\) scores are to be summed. The index variable \(i\) goes from \(1\) to \(3\).

    When all the scores of a variable (such as \(X\)) are to be summed, it is often convenient to use the following abbreviated notation:

    \[ \sum X \]

    Thus, when no values of i are shown, it means to sum all the values of \(X\).

    Many formulas involve squaring numbers before they are summed. This is indicated as

    \[ \sum X^2 = 4.62 + 5.12 + 4.92 + 4.42 = 21.16 + 26.01 + 24.01 + 19.36 = 90.54 \]

    Notice that:

    \[ \left(\sum X \right)^2 \neq \sum X^2 \]

    because the expression on the left means to sum up all the values of \(X\) and then square the sum (\(19^2 = 361\)), whereas the expression on the right means to square the numbers and then sum the squares (\(90.54\), as shown).

    Some formulas involve the sum of cross products. Table \(\PageIndex{2}\) shows the data for variables \(X\) and \(Y\). The cross products (\(XY\)) are shown in the third column. The sum of the cross products is \(3+4+21 = 28\).

    Table \(\PageIndex{2}\): Cross Products.
    X Y XY
    1 3 3
    2 2 4
    3 7 21

    In summation notation, this is written as:

    \[\sum XY = 28.\]

    Contributors and Attributions

    • Online Statistics Education: A Multimedia Course of Study ( Project Leader: David M. Lane, Rice University.

    • David M. Lane
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