MATH LECTURE 4.1
SUMMATION PROCESS using SIGMA NOTATION
The study of statistics involves the collection of data or measurement. Thus, there is always a need to add
several numbers. The Greek capital letter sigma, 𝜮 is used in the process. The symbol 𝚺�, read as “the sum” of
tells you to add certain numerical values.
SIGMA NOTATION
where:
𝑛
𝛴
𝑥
𝑖
𝑛
∑𝑥
𝑖
signifies summation process.
expressions to be sum up.
lower limit (or starting value of 𝑥)
upper limit (or ending value of 𝑥)
Illustrative Examples:
1. ∑51(𝑥) = 1 + 2 + 3 + 4 + 5
= 𝟏𝟓
“The starting value of 𝑥 is 1 and ends in 5. Thus 𝑥 = {1, 2, 3, 4, 5}.
2. ∑62(𝑥 2 ) = 22 + 32 + 42 + 52 + 62
= 4 + 9 + 16 + 25 + 36
= 𝟗𝟎
“The starting value of 𝑥 is 1 and ends in 5. Thus 𝑥 = {2, 3, 4, 5, 6}.
3. ∑81(5) = 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5
= 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8
= 𝟒𝟎
“If no variable is found, the upper limit represents
the number of terms in the equation”
Sigma Notation is used to shorten the summation expression of sequence, whether Arithmetic or Geometric.
Illustrative Examples:
Express the sum of numbers in Sigma Notation.
1. 1 + 2 + 3 + ⋯ + 98 + 99 + 100
let 𝒙 represents 1, 2, 3, 4, …, 98, 99, and 100, where 1 is the lower limit (𝑖 = 1) and 100 the upper limit (𝑛 =
100). Since the expression show an increment of 1 (or each term is increasing by 1), therefore,
100
∑
1
𝟏𝟎𝟎
𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥98 + 𝑥99 + 𝑥100 = ∑
𝒙
𝟏
2. 3 + 5 + 7 + 9
There are FOUR TERMS in the expression. Let 𝒙 represents 1, 2, 3, and 4, where 1 is the lower limit (𝑖 = 1)
and 4 the upper limit (𝑛 = 4), the pattern shows an increment of 2 for each term and all terms are ODD
numbers, therefore,
4
∑ (2𝑥 + 1)
1
3. 2 + 4 + 8 + 16 + 32
There are FIVE TERMS in the expression. Let 𝒙 represents 1, 2, 3, 4, and 5, where 1 is the lower limit (𝑖 = 1)
and 5 the upper limit (𝑛 = 5), the pattern shows an exponential increase of each term and the first term is 2,
therefore,
5
∑ 2𝑥
1
4. Consider the scores obtained by 10 students in a 50-items mathematics test.
For convenience, variables will be used to present the data.
Student No.
1
2
3
4
5
6
7
8
9
10
Score
35
40
29
37
25
33
49
47
28
42
Let
𝑥 = score obtained by each student
𝑥𝑖 = different values or observations of 𝑥
𝑥𝑖 is read as “𝑥 sub 𝑖” where 𝑖 is a subscript which indicates the position of
each value in the series.
In the given data, there are 10 observations denoted as
𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9 , 𝑥10 .
Hence,
10
∑ 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 + 𝑥6 + 𝑥7 + 𝑥8 + 𝑥9 + 𝑥10
𝑖=1
The symbol
10
∑ 𝑥𝑖
𝑖=1
is read as “the sum of 10 observations 𝑥1 to 𝑥10 ”. To substitute the data:
10
∑ 𝑥𝑖 = 35 + 40 + 29 + 37 + 25 + 33 + 49 + 47 + 28 + 42 = 𝟑𝟔𝟓
𝑖=1
5. Given are the ages of the first 4 shoppers at a newly opened convenience store in the neighborhood.
12, 24, 30, 45.
Guide Questions:
a. What will 𝒙 represent in the information given?
Answer:
𝒙 will represent the ages of the first 4 shoppers in the newly opened convenience
store.
b. What will the subscript 𝒊 represent?
Answer:
𝒊 will represent the first 4 shoppers in the newly opened convenience store
c. Write an expression for the sum.
∑41 𝑥𝑖
Answer:
d. What are the lower and upper limits of the expression?
Answer:
The lower limit is 1 and the upper limit is 4.
e. Write the formula for the summation and find the sum of the given information.
∑41 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4
Answer:
= 12 + 24 + 30 + 45
= 𝟏𝟏𝟏
This time, consider 5 observations. If the sum of five observations is written as:
5
∑ 𝑥𝑖 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5
1
the sum of the squares of the five observations is represented as:
5
∑ 𝑥𝑖2 = 𝑥12 + 𝑥22 + 𝑥32 + 𝑥42 + 𝑥52
1
the sum of the products of pairs of five observations is expressed as:
5
∑ 𝑎𝑖 𝑥𝑖 = 𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + 𝑎4 𝑥4 + 𝑎5 𝑥5
5
1
∑ 𝑎𝑖 𝑥𝑖 = (1)𝑥1 + (2)𝑥2 + (3)𝑥3 + (4)𝑥4 + (5)𝑥5
1
END OF MATH LECTURE 4.1