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# Summation Notation

Module by: David Lane. E-mail the author

Summary: (Blank Abstract)

## Summation Notation

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 XX that represents the weights (in grams) of 44 grapes. The data are shown in Table 1.

Table 1: Weights of 4 grapes
Grape X
1 4.6
2 5.1
3 4.9
4 4.4

We label Grape 11's weight X 1 X 1 , Grape 22's weight X 2 X 2 , etc. The following formula means to sum up the weights of the four grapes: i =14 X i i 1 4 X i The Greek letter ΣΣ indicates summation. The " i=1 i 1 " at the bottom indicates that the summation is to start with X 1 X 1 and the 44 at the top indicates that the summation will end with X 4 X 4 . The X i X i indicates that XX is the variable to be summed as ii goes from 11 to 44. Therefore, i =14 X i = X 1 + X 2 + X 3 + X 4 =4.6+5.1+4.9+4.4=19.0 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 i =13 X i i 1 3 X i indicates that only the first 33 scores are to be summed. The index variable ii goes from 11 to 33. When all the scores of a variable (such as XX) are to be summed, it is often convenient to use the following abbreviated notation: X X Thus when no values of ii are shown, it means to sum all the values of XX.

Many formulas involve squaring numbers before they are summed. This is indicated as X2=4.62+5.12+4.92+4.42=21.16+26.01+24.01+19.36=90.54 X 2 4.6 2 5.1 2 4.9 2 4.4 2 21.16 26.01 24.01 19.36 90.54 Notice that: X2X2 X 2 X 2 because the expression on the left means to sum up all the values of XX and then square the sum ( 192=381 19 2 381 ) whereas the expression on the right means to square the numbers and then sum the squares ( 90.5490.54, as shown).

Some formulas involve the sum of cross products. Table 2 shows the data for variables XX and YY. The totals are shown in the bottom row and the cross products are shown in the third column. The sum of the cross products is 2828.

Table 2: Sum of Cross Products
X Y XY
1 3 3
2 2 4
3 7 21
6 12 28

In summation notation, this is written as: XY=28 X Y 28

## Self Test Questions

For the following data

Table 3
X Y
1 3
2 2
4 7

compute X X

1+2+4=7 1 2 4 7

### Exercise 2

compute Y Y

#### Solution

3+2+7=12 3 2 7 12

### Exercise 3

compute X2 X 2

#### Solution

1+2+4=7 1 2 4 7 and 7×7=49 7 7 49

### Exercise 4

compute i =23 X i i 2 3 X i

#### Solution

This means to add the values of XX from the second value to the third value. 2+4=6 2 4 6 .

### Exercise 5

compute XY X Y

#### Solution

1×3+2×2+4×7=3+4+28=35 1 3 2 2 4 7 3 4 28 35

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