Summation

Introduction to Summation Notation

• Many statistical formulas involve summing numbers.

• There exists a convenient notation for expressing summation.

Basic Notation of Summation

• The variable x represents weights in grams of four grapes.

  • Grape one's weight is referred to as x₁.

  • Grape two's weight is referred to as x₂.

  • Grape three's weight is referred to as x₃.

  • Grape four's weight is referred to as x₄.

Summation Expression

• The sum of the four weights is expressed in summation notation.

  • The Greek letter sigma (Σ) indicates the operation of summation.

  • The notation Σ indicates:

    • Bottom Index (I = 1): The summation starts with the first variable (x₁).

    • Top Index (4): The summation ends with the fourth variable (x₄).

• Therefore, the summation notation can be interpreted as:

  • $\Sigma<em>{i=1}^{4} x</em>{i}$

    • This means to sum x₁, x₂, x₃, and x₄.

  • The actual weights are:

    • x₁ = 4.6

    • x₂ = 5.1

    • x₃ = 4.9

    • x₄ = 4.4

Calculation of the Sum

• The total sum of the weights of the grapes:

  • $4.6 + 5.1 + 4.9 + 4.4 = 19$

Specific Summation Notation

• A different expression can sum only the first three values.

  • $\Sigma<em>{i=1}^{3} x</em>{i}$

    • Index variable I goes from 1 to 3.

• Alternatively, when no index values are shown, it indicates the sum of all values:

  • $\Sigma x_{i}$ implies summing all x values.

Squaring Values Before Summation

• It is important to note that many formulas involve squaring numbers before they are summed.

  • This notation indicates:

    • $\Sigma (x_{i})^2$

      • Each value of x is squared first and then summed.

  • It matters if you sum the values first and then square or square first and then sum.

    • Example of squaring the sum:

      • $(\Sigma x_{i})^2$

      • This expression equates to:

        • $19^2 = 361$

    • Example for summing squares:

      • $\Sigma (x_{i})^2$ produces:

        • For example, the sum of squares could be $90.54$.

Application of Cross Products in Summation

• Some formulas also involve the sum of cross products.

  • A table showing data for two variables, X and Y, is used.

    • The third column represents the cross products (x times y).

• The sum of the cross products for these variables is:

  • Total = 28

  • The corresponding formula using summation notation is:

    • $\Sigma(x<em>{i} * y</em>{i})$

    • This denotes the total of all cross-products of x and y.