Lab 1: Summation Notation and Order of Operations Study Guide

Introduction to Summation Notation

• Definition of Notation: Notation is simply the stylized way in which mathematical concepts are written down. It is a specific format for communicating mathematical operations.
• Definition of Summation: Summation refers to the act of addition, specifically adding up all values within a given set. It implies adding all the numbers you have available, rather than just a selection of them.
• The Sigma Symbol (\Sigma):     * The Greek letter used to denote summation is the capital letter Sigma ($\Sigma$).     * In a mathematical expression, the presence of $\Sigma$ serves as an instruction to "add up all that follows."     * Lowercase Sigma ($\sigma$): While the uppercase version ($\Sigma$) is for summation, the lowercase version ($\sigma$) is used for entirely different concepts later in the statistics course.

Statistical Variables and Sample Size

• Variable Designation: Variables in statistics are typically represented by letters, most commonly $x$ and $y$.     * If you measure 30 people on a variable $x$, you will have 30 individual scores for that variable.     * The expression $\sum x$ tells the researcher to add up all 30 of those scores.
• The Variable "$n$/$N$": This letter represents the number of individuals or observations.     * Big $N$ ($N$): This represents the number of individuals within a population.     * Small $n$ ($n$): This is a critical value in statistics known as the sample size. It refers to the number of individuals within a specific sample.

Order of Operations: PEMDAS Review

To correctly evaluate summation expressions, one must follow the standard order of operations, often remembered by the acronym PEMDAS.

• $P$ (Parentheses): Perform any operations inside parentheses first. This is a mechanism used to force a specific operation to occur out of its normal natural order.
• $E$ (Exponents): Perform squaring ($x^2$) or square roots. In this specific coursework, squaring is the primary exponent used. Square roots are technically exponents of one-half ($\frac{1}{2}$).
• $M$ and $D$ (Multiply and Divide): These operations are performed as a single unit, moving from left to right through the expression.
• $A$ and $S$ (Add and Subtract): These operations are also performed as a unit from left to right. Since summation ($\Sigma$) is a form of addition, it falls into this category.

Step-by-Step Problem Solving: Single Variable

Using the scores set: $4, 3, 7, 1$ (where population size is $n = 4$ and the variable is $x$).

• Problem 1a: Calculate $\sum x$     * This is the simplest form, requiring only addition.     * Calculation: $4 + 3 + 7 + 1 = 15$.     * Result: $15$.

• Problem 1b: Calculate $\sum x^2$     * Operations: Addition ($\Sigma$) and an Exponent (squaring).     * Rule: Exponents come before addition in PEMDAS.     * Procedure: Because there is no existing sum yet, you square each individual score first to create a new column of data.     * Calculations:         * $4^2 = 16$         * $3^2 = 9$         * $7^2 = 49$         * $1^2 = 1$     * Summing the squares: $16 + 9 + 49 + 1 = 75$.     * Result: $75$.

• Problem 1c: Calculate $(\sum x)^2$     * Operations: Summation within parentheses and a square outside.     * Rule: Parentheses take priority ($P$ before $E$).     * Procedure: Sum the $x$ values first, then square that total sum.     * Calculations: $\sum x = 15$. Then, square the result: $15^2 = 225$.     * Result: $225$.

• Problem 1d: Calculate $\sum x - 1$     * Operations: Addition ($\Sigma$) and Subtraction.     * Rule: Add and subtract from left to right. Summation typically appears as the leftmost operation.     * Procedure: Sum the scores first to get a total, then subtract 1 from that total.     * Calculations: $\sum x = 15$. Then, $15 - 1 = 14$.     * Result: $14$.

• Problem 1e: Calculate $\sum (x - 1)$     * Operations: Summation and subtraction within parentheses.     * Rule: Parentheses take priority.     * Procedure: Since we do not have a sum yet, perform the subtraction on every individual score first. Then, sum the results.     * Calculations:         * $4 - 1 = 3$         * $3 - 1 = 2$         * $7 - 1 = 6$         * $1 - 1 = 0$     * Summing the new values: $3 + 2 + 6 + 0 = 11$.     * Result: $11$.

• Problem 1f: Calculate $\sum (x - 1)^2$     * Operations: Addition ($\Sigma$), Subtraction (in parentheses), and Exponent (squaring).     * Rule: Parentheses $>$ Exponents $>$ Summation.     * Step 1 (Parentheses): Subtract 1 from every score to get $3, 2, 6, 0$.     * Step 2 (Exponent): Square each of those results to get $3^2=9, 2^2=4, 6^2=36, 0^2=0$.     * Step 3 (Summation): Add the squared results together: $9 + 4 + 36 + 0 = 49$.     * Result: $49$.

Identification of Mathematical First Steps

• Example A: $\sum x^2$     * The first step is squaring (Exponents), which occurs before adding (Summation).
• Example B: $(\sum x)^2$     * The first step is adding up the $x$ values because they are contained within parentheses.
• Example C: $\sum (x - 2)^2$     * The first step is subtraction ($x - 2$) because it is inside the parentheses. Since no sum exists yet, this is performed on every individual score.

Expressing Verbal Instructions as Summation Notation

• Instruction 3a: "Subtract 2 points from each score, then add the resulting values."     * Translation: The subtraction must happen first, necessitating parentheses: $\sum (x - 2)$.
• Instruction 3b: "Subtract 2 points from each score, square the resulting values, then add the squared numbers."     * Translation: Parentheses force the subtraction ($x-2$), the square resides outside the parentheses, and the summation symbol applies to the final results: $\sum (x - 2)^2$.
• Instruction 3c: "Add the scores and then square the total."     * Translation: To force the addition to happen before the exponent, the summation must be in parentheses: $(\sum x)^2$.

Calculations with Multiple Variables ($x$ and $y$)

In scenarios involving two variables, scores often come in pairs (e.g., height as variable $x$ and weight as variable $y$ for the same individual).

• Data Set Example:     * Variable $x: 1, 2, 2, 4$     * Variable $y: 2, 3, 7, 6$
• Evaluating $\sum xy$:     * Operations: Multiplications and Summation.     * Rule: Multiplication (adjacent variables indicate multiplication) happens before addition ($\Sigma$).     * Procedure: Multiply the pairs first to create a new column of products.     * Calculations of Products ($xy$):         * $1 \times 2 = 2$         * $2 \times 3 = 6$         * $2 \times 7 = 14$         * $4 \times 6 = 24$     * Summing the products: $2 + 6 + 14 + 24 = 46$.     * Final Result: $46$.