1.1 Notating Summation

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Contributor:

Erica Madebeykin
PGDip Researcher

Creation Date: March 12, 2024

❤️ Reference Acknowledgments: London School of Economics (world rank#45)

This review reflects the author's learning and thoughts based on the materials covered in the FREE online course STAT104a provided by the London School of Economics, delivered on the EdX platform.
A sincere thank you to Professor James Abdey for delivering such an amazing course!

Capital Sigma letter: The Summation Operator

In mathematical notations, you will encounter quite often this symbol (Σ):

Mathematical notations with Sigma (Σ)

$$\sum _{i=1}^n{x}_i$$

You will quickly notice the Σ symbol in formulas for calculating statistics like the mean and standard deviation:

$$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}{s}_x=\sqrt{\frac{\sum _{i=1}^n(x_i-\overline{x}{)}^2}{n-1}}$$

As you progress into inferential statistics, you'll might encounter this as well:
One-way ANOVA (Sums of squares for y) formula

$$\sum _{j=1}^g\sum _{i=1}^{n_g}(y_{ij}-\overline{y}{)}^2=\sum _{j=1}^g{n}_j({\overline{y}}_j-\overline{y}{)}^2+\sum _{j=1}^g\sum _{i=1}^{n_g}(y_{ij}-\overline{y_j}{)}^2$$

So what exactly does this symbol (Σ) do within mathematical equations?

Step by Step Explanations:

• Capital Greek Letter Sigma
  • is used as a shorthand mathematical notation for the Summation Operator or "sum of values observed"
    
    
• Index and Range of Summation
  • Range :
  • When we talk about the "Index and Range of Summation," it's like counting steps when adding things up. Imagine you're adding numbers together, starting from a certain point (let's call it "i") and going up to a certain number (let's call it "n").
    
    
  • 2 Interpretations of small letter n above the capital sigma symbol (Σ):
  • Default Interpretation: When the small letter "n" appears above the sigma symbol (Σ) without a specific number, it signifies the total number of elements in a set of values.
  • Indexed Interpretation: When a specific number replaces "n" above the sigma symbol, it represents the last index inclusive for counting, when summing up the values.
    
    
  • Indexing :
  • The "i=1" under the symbol (∑) tells us where to start counting from. In this case, we start at 1, but sometimes we might start at a different number.
  • The "i" is a placeholder that tells us which number we're adding at the moment. It starts at 1 and goes up to whatever the value of "n" is.
  • We can change the letter "i" to something else if we want, like "j" or "k." It's just a way to keep track of the steps as we add things together.:

$$\sum _{j=1}^n{x}_j\sum _{k=1}^n{x}_k$$

IMPORTANT NOTE!
Mathematicians start their indexing from a minimum value of 1.
In contrast, the Programmers start from a minimum value of 0.

$$\sum _{i=1}^n{x}_i\sum _{i=0}^n{x}_i$$

The symbol "x"

• The symbol "x" next to the Greek letter sigma (Σ) is a placeholder variable that can stand for any value, like a number. It's flexible and can represent different things, like whole numbers or decimals.
• The letter x has a subscript i which implies the "order" of the x-variable within the set of observations
• This subscript "i" depends on the initializer (i) and stopper values (n) pre-declared in the sigma.
• Therefore, the set of observed "x" values can be interpreted as:

$$x_i=1,2,...,n\{x_i\mid i=1,2,\dots ,n\}$$

• When the sigma or summation operator is attached before the set of x observations, it means that all the x-values should be added together.
• The variable "x" also often goes together with another variable, usually "y," which helps create a graph called the Cartesian plane.

$$y_i=1,2,...,n\{y_i\mid i=1,2,\dots ,n\}$$

Introduction to set of values

Let's jump into the math excitement from LSE's Stat104a course! This free online class on EdX demonstrates how Sigma is being used in actual calculations.

First, let's check out the sample dataset that we will use, as discussed in the course:

$$\{x_i\mid i=4,5,1,-2,9\}$$

In short, what does this set of values tells you?

Given this data set, the numbers 4, 5, 1, -2, 9 tells you that:

$$x_1=4,x_2=5,x_3=1,x_4=-2,x_5=9$$$$n=5$$

How do you read that?

You can read the above mathematical notation as:
"The set of x values labeled by i, with i taking on the values 4, 5, 1, -2, and 9 "

Further explanation
This notation defines a set containing of x values, where each x value is associated with a specific i value. In this case, the i values are 4, 5, 1, -2, and 9.

Reminder: For set of values, "n" means the total number of elements in your set.

Practice Calculations using Sigma

In the same LSE's Stat104a course, the following calculations were performed on the previous set of values using the Sigma Operator. You can try to solve the equations yourself, and then verify your answers by clicking the solutions.

Given the set of values:

$$\{x_i\mid i=4,5,1,-2,9\}$$

Practice Task #1

Calculate:

$$\sum _{i=4}^n{x}_i$$

Solution: Click here to see the solution

$$\sum _{i=4}^n{x}_i=\sum _4^5{x}_4+x_5$$$$=-2+9=7$$

Practice Task #2

Calculate:

$$\sum _{i=1}^4{x}_i^2$$

Solution: Click here to see the solution

$$\sum _{i=1}^4{x}_i^2=x_1^2+x_2^2+x_3^2+x_4^2$$$$=4^2+5^2+1^2+2^2$$$$=16+25+1+4$$$$=46$$

Practice Task #3

Calculate:

$$\sum _{i=4}^5{x}_i(x_i-2)$$

Solution: Click here to see the solution

$$\sum _{i=4}^5{x}_i(x_i-2)$$$$=\sum _{i=4}^5{x}_i^2-2x_i$$$$=(x_4^2-2\cdot x_4)+(x_5^2-2\cdot x_5)$$$$=(-2^2-(2\ast -2))+(9^2-(2\ast 9))$$$$=(4--4)+(81-18)$$$$=8+63$$$$=71$$

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