09-25-2025 P04 FinAlg Sigma Intro

By Antoinette Norris Woodson

​Agenda
Bell Ringer/Poll Question
Objectives
Knowledge Check
New Stuff! 😊

Poll

U rock!

How are you doing on a sale of 1 - 5 today?

• 1

• 2

• 3

• 4

• 5

Poll

Do you know what sigma is/means?

• yes

• no

Discretionary and Essential Expenses

Objectives

•Identify the difference between essential and discretionary expenses.

•Determine the mean, median, and mode of a data set.

•Use sigma notation to represent and determine the mean of a data set.
•Create and interpret a frequency distribution table.

•Determine the mean, median, and mode of a data set presented in a frequency distribution table.

•median

•mode

•subscript

•index

•outlier
Sigma

•skewed data set

•bimodal

•frequency distribution

•gross income

•disposable income

•essential expense

•discretionary expense

•statistics

•data

•measures of central tendency

•mean

Key Terms

​Take out a piece of paper/pen or pencil NO SCREENSHOTS

Greek letter that represents a sum in mathematics.
It is a concise way of representing a series.

Using Σ to compute a mean more cleanly.
It means "sum of"

The Formula for the Mean using Sigma

Mean via Sigma notation

​x̄ = the mean (average)
Σ = the sum of xᵢ = ith data value
(like the 1st, 2nd, 3rd number…)
i = index (starts at 1, goes to n)
n = number of data points

​No screenshots
Write it down

Why Use Sigma Notation?

• Writing out all the numbers (27 + 31 + 30 + 26 + … + 26) is long and messy.

• Sigma (Σ) is a shortcut symbol that means “add everything up.”

• This saves space and makes the formula for the mean cleaner:

• igma notation becomes even more powerful with large data sets (like 100 test scores or 365 daily temperatures).

• It’s a universal math language — every statistic formula (mean, variance, standard deviation) relies on sigma notation.

Regular Method

Step 1 – Add them all:
128 + 118 + 96 + 102 + 100 + 118 + 118 + 102

= 882

Step 2 – Count how many numbers:
n = 8

Step 3 – Divide:
Mean = 882 ÷ 8 = 110.25

Let's Compare

Nora is a college student. She needs to make an essential textbook purchase for one of her classes. She researches the cost of the book at her college bookstore, some local booksellers, and some online merchants. The prices per book are $128, $118, $96, $102, $100, $118, $118, and $102. Find the mean of the textbook prices.

Sigma Method
x̄ = the mean (average)

Σ = the sum of all the textbook prices

xᵢ = ith data value:
x₁ = 128, x₂ = 118, x₃ = 96, x₄ = 102, x₅ = 100, x₆ = 118, x₇ = 118, x₈ = 102

i = index, starts at 1 and ends at 8

n = number of data points, here n = 8

Let's Compare

Abby's Water Bills

Addy’s monthly water bills for last year are $27, $31, $30, $26, $25, $27, $37, $33, $32, $28, $26, $26.


Express the formula for the mean using sigma notation and calculate the mean water bill for the year.

Abby's Water Bills

Addy’s monthly water bills for last year are $27, $31, $30, $26, $25, $27, $37, $33, $32, $28, $26, $26.

Express the formula for the mean using sigma notation and calculate the mean water bill for the year.
QUESTIONS

• How many values are there? (What is n?)

• What is the sum of all the values? (What is Σxi\Sigma x_iΣxi​?)

• What does each xix_ixi​ represent?

• How would you write the formula for the mean using sigma notation?

Abby's Water Bills-$27, $31, $30, $26, $25, $27, $37, $33, $32, $28, $26, $26.

Q: How many values are there? (What is n?)
A: There are 12 values, so n = 12.

Q: What is the sum of all the values? (What is Σxᵢ?)
A: Σxᵢ = 348

Q: What does each xᵢ represent?
A: Each xᵢ is one monthly water bill (e.g., x₁ = 27, x₂ = 31, …, x₁₂ = 26).

Abby's Water Bills-$27, $31, $30, $26, $25, $27, $37, $33, $32, $28, $26, $26.

​Calcuations in Calc

​Carlos tracked his grocery bills for 6 weeks: $82, $96, $75, $90, $87, $100. Express the formula for the mean using sigma notation and calculate the mean grocery bill.

​Carlos’s grocery bills for 6 weeks were $82, $96, $75, $90, $87, $100. Express the formula for the mean using sigma notation and calculate the mean grocery bill.

QUESTIONS How many values are there? (What is 𝑛 n?)

What is the sum of all the values? (What is Σ 𝑥 𝑖 Σx i ​ ?)

What does each 𝑥 𝑖 x i ​ represent?

How would you write the formula for the mean using sigma notation?

​Q: How many values are there? (What is n?)
A: There are 6 values, so 𝑛 = 6 n=6.

Q: What is the sum of all the values? (What is Σxᵢ?)
A: Σ 𝑥 𝑖 = 82 + 96 + 75 + 90 + 87 + 100 = 530 Σx i ​ =82+96+75+90+87+100=530.

Q: What does each xᵢ represent?
A: Each 𝑥 𝑖 x i ​ is one weekly grocery bill (e.g., 𝑥 1 = 82 ,    𝑥 2 = 96 ,    … ,    𝑥 6 = 100 x 1 ​ =82,x 2 ​ =96,…,x 6 ​ =100).

Sigma Notation Solution

​Answer: The mean weekly grocery bill is $88.33.

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