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Week 3 - Summation Notation

The document provides an overview of summation notation and its application in calculating measures of central tendency. It introduces the notation for summation, presents various theorems related to summation, and includes examples and assignments for practice. The content is structured to help readers understand how to express and manipulate summation notation effectively.

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Les Lie
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12 views9 pages

Week 3 - Summation Notation

The document provides an overview of summation notation and its application in calculating measures of central tendency. It introduces the notation for summation, presents various theorems related to summation, and includes examples and assignments for practice. The content is structured to help readers understand how to express and manipulate summation notation effectively.

Uploaded by

Les Lie
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 AND

MEASURES OF CENTRAL
TENDENCY
Introduction: Notation and Average
SUMMATION NOTATION
An Overview
N O TAT I O N S A N D S Y M B O L S
Suppose that a variable X is the variable of interest, and that n
measurements are taken. The notation 𝑋1 , 𝑋2 , … , 𝑋𝑛 will be used to
represent the n observation.
Let the Greek letter Σ indicate the “summation of”, thus, we can
write the sum of n observation as
𝑛

෍ 𝑋𝑖 = 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛 .
𝑖=1
I L L U S T R AT I O N S : E X PA N D T H E
F O L L O W I N G S U M M AT I O N N O TAT I O N
1. σ5𝑖=1 𝑋𝑖 6. σ3𝑎=1 𝑌𝑎 2

σ6
2. σ1000
𝑗=2 𝑌𝑗
7. 𝑖=4 𝑋𝑖 + 𝑌𝑖
3. σ10
𝑘=2 𝑍 𝑘 8. σ15
𝑖=10 𝑋𝑖 𝑌𝑖
3 2 𝐴𝑖
σ
4. 𝑎=1 𝑌𝑎 9. 8
σ𝑖=1
𝐵𝑖
σ5
5. 𝑖=1 2𝐴𝑖 3
10. 𝑖=1 𝐷 𝑖+2
σ
I L L U S T R AT I O N S : E X P R E S S T H E
F O L L O W I N G I N S U M M AT I O N N O TAT I O N
1. 𝑋1 + 𝑋2 + 𝑋3 + ⋯ + 𝑋50
𝐴6 3 +𝐴7 3 +𝐴8 3
2. 5

3. (𝐾2 + 𝐾3 )(𝑃4 + 𝑃5 )
4. 1 + 2 + 3 + ⋯ + 1000
5. 𝑍3 + 𝑍5 + 𝑍7 + ⋯ + 𝑍13
T H E O R E M S O N S U M M AT I O N
N O TAT I O N
1. If c is a constant, then
𝑛

෍ 𝑐 = 𝑛𝑐
𝑖=1

2. If c is a constant, then
𝑛 𝑛

෍ 𝑐𝑥𝑖 = 𝑐 ෍ 𝑥𝑖
𝑖=1 𝑖=1
T H E O R E M S O N S U M M AT I O N
N O TAT I O N ( C O N T I N U AT I O N )
3. The summation of the sum of two or more variable is the sum of their
summations. Thus,
𝑛 𝑛 𝑛

෍(𝑥𝑖 ±𝑦𝑖 ) = ෍ 𝑥𝑖 ± ෍ 𝑦𝑖
𝑖=1 𝑖=1 𝑖=1
4. The summation of a variable and a constant is the sum of their
summations. Thus,
𝑛 𝑛

෍(𝑥𝑖 ± 𝑐) = ෍ 𝑥𝑖 ± 𝑛𝑐
𝑖=1 𝑖=1
T H E O R E M S O N S U M M AT I O N
N O TAT I O N ( C O N T I N U AT I O N )
5. σ𝑛𝑖=1(𝑥𝑖 ± 𝑐)2 = σ𝑛𝑖=1 𝑥𝑖 2 ± 2𝑐 σ𝑛𝑖=1 𝑥𝑖 + 𝑛𝑐 2
6. If n is a positive integer, then
𝑛(𝑛+1)
a. σ𝑛𝑖=1 𝑖 = 2
𝑛 2 𝑛(2𝑛+1)(𝑛+1)
b. σ𝑖=1 𝑖 =
6
𝑛2 (𝑛+1)2
c. σ𝑛𝑖=1 𝑖 3 = 4
𝑛 𝑛+1 2𝑛+1 (3𝑛2 +3𝑛−1)
d. σ𝑛𝑖=1 𝑖 4 = 30
A S S I G N M E N T : E VA L U AT E T H E
F O L LOW I N G B Y A P P LY I N G T H E O R E M S O F
S U M M AT I O N N O TAT I O N
The number of computer units (in thousands) manufactured by XYZ
Company is given as follows : 𝑋1 = 10; 𝑋2 = 12; 𝑋3 = 14; 𝑋4 =
10; and 𝑋5 = 15. Determine the exact value of the following,
a. σ5𝑖=1 𝑋𝑖
b. σ3𝑖=1 2𝑋𝑖
c. σ5𝑖=3 𝑋𝑖 − 4

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