Skip to content Skip to navigation

Connexions

You are here: Home » Content » Summation Notation

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Summation Notation

Module by: David Lane. E-mail the author

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: (Blank Abstract)

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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: ∑X2≠∑X2 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

Exercise 1

compute ∑X X

Solution

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

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks