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Lesson 2.1 Summation Notation

The document discusses summation notation and how it can be used to compactly represent the sum of values. Examples are provided to demonstrate how to evaluate summations by replacing the index variable with values within the specified range and adding the results.

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Zappy Leoj A. Lopez
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54 views14 pages

Lesson 2.1 Summation Notation

The document discusses summation notation and how it can be used to compactly represent the sum of values. Examples are provided to demonstrate how to evaluate summations by replacing the index variable with values within the specified range and adding the results.

Uploaded by

Zappy Leoj A. Lopez
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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SUMMATION

NOTATION
“ Most of the calculations
we perform in statistics
are repetitive operations
on lists of numbers.

2
 Sigma
▸ Sigma notation is used to denote the
sum of all values.

▸ It is also called the summation


notation.

3
For example, suppose
we weigh five children.
We will denote their weights by 𝑥1 , 𝑥2 , 𝑥3, 𝑥4 and 𝑥5

The sum of their weights can be written as 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5

The sum can be expressed more compactly using sigma as

𝑥𝑖
𝑖=1

4
Thus,
5

𝑥𝑖 =
𝑖=1

The symbol ∑ means “add up”. Underneath


∑ we see 𝑖 = 1 and on top of it 5. This
means that 𝑖 is replaced by whole numbers
starting at the bottom number, 1, until the
top number, 5, is reached.

5
Now, let us find
4
where 𝑥1 = 2, 𝑥2 = 3, 𝑥3 = −2, and 𝑥4 = 1.
2𝑥𝑖
𝑖=1
4

2𝑥𝑖 = 2𝑥1 + 2𝑥2 + 2𝑥3 + 2𝑥4


𝑖=1
= 2 2 + 2 3 + 2 −2 + 2(1)

= 4 + 6 + −4 + 2

= 8
6
Similarly, let us find
3

( 𝑥𝑖 − 4) where 𝑥1 = 7, 𝑥2 = 4, 𝑥3 = 1.
𝑖=1

( 𝒙𝒊 − 𝟒) = (𝒙𝟏 − 𝟒) + (𝒙𝟐 − 𝟒) + (𝒙𝟑 − 𝟒) = 𝟕 − 𝟒 + 𝟒 − 𝟒 + (𝟏 − 𝟒) = 𝟎


𝒊=𝟏

7
Notice that is
different from
𝟑

𝒙𝒊 − 𝟒 where 𝑥1 = 7, 𝑥2 = 4, 𝑥3 = 1.
𝒊=𝟏

𝑥𝑖 − 4 = 𝑥1 + 𝑥2 + 𝑥3 − 4
𝑖=1
=7+4+1−4

= 8

8
3

Write in full. 𝑖=1


𝑥𝑖

𝑥 𝑖 = 𝑥 1 +𝑥 2 + 𝑥 3
𝑖=1

9
We also use sigma in the following way.
4

𝑖2
𝑖=1

Replace 𝑖 in the expression


(this time 𝑖 2 ) by whole
numbers starting with 1 and
ending with 4, and add.
4

𝑖 2 = 12 +22 + 32 + 42 = 30
𝑖=1
10
Your Turn
1. Let 𝑥1 = 8, 𝑥2 = 9, 𝑥3 = 12, 𝑥4 = 15, 𝑥5 = 6, 𝑥6 = 3, 𝑥7 = 10, 𝑥8 = 5, 𝑥9 = 2, 𝑥10 = 1.
10

a. Evaluate 𝑥𝑖
𝑖=1

b. Evaluate 𝑥𝑖
𝑖=1

c. Evaluate 𝑥𝑖
𝑖=2

11
Your Turn
4

2. Evaluate 𝑥𝑖 where 𝑥1 = 5, 𝑥2 = 2, 𝑥3 = 3, 𝑥4 = 8.
𝑖=1

3. Evaluate 5𝑥𝑖 where 𝑥1 = 10, 𝑥2 = 14, 𝑥3 = −2.


𝑖=1

4. Find the value of (𝑥𝑖 −2)2 where 𝑥1 = 105, 𝑥2 = 100, 𝑥3 = 95.


𝑖=1

5. Find the value of 𝑖+2


𝑖=1 12
“ The whole is greater
than the sum of its
parts. - Aristotle
13
Assignment 2.1
5
1 where 𝑥1 = 10 kg, 𝑥2 = 12 kg, 𝑥3 = 14 kg, 𝑥4 = 8 kg and 𝑥5 = 11 kg
a. Find 𝜇= 𝑥
5 𝑖
𝑖=1 are the weights of 5 children. (𝜇 is the mean weight of the children)

b. Let 𝑥1 = 8, 𝑥2 = 9, 𝑥3 = 12, 𝑥4 = 15, 𝑥5 = 6, 𝑥6 = 3, 𝑥7 = 10, 𝑥8 = 5, 𝑥9 = 2, 𝑥10 = 1.

Evaluate 8𝑥𝑖
𝑖=4
3

c. Find the value of (2𝑖 + 1)2


𝑖=1

d. Evaluate 3𝑘 − 5
𝑘=3

14

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