Topic 0 Math Skills Review

Variables

Definition: A variable is an entity or characteristic that can take on different values among subjects.
Examples:
- Biological Sex
- Rank in class
- Clothing sizes (S, M, L, XL)
- Temperature
- Life Satisfaction Score
- Number of correct answers
- Time to complete a task
- Gain in height since last year
- Depression Score

Discrete and Continuous Variables

Discrete Variable:
- Has separate categories.
- No values can exist between two neighboring categories.
- Examples: Biological Sex, Number of students in the classroom.
Continuous Variable:
- There are an infinite number of possible values between any two observed values.
- The variable can take on any value at any point along an interval.
- Example: Height.

Levels of Measurement

Stevens (1946) classified variables into four levels, often referred to as levels of measurement or levels of data. Each subsequent level possesses the characteristics of the preceding ones, with additional properties.

Nominal Scale:
- Description: Attributes are only named; the weakest level.
- Characteristics:
  - Just names (simply label objects).
  - Cannot be ordered.
  - Non-numeric values (though values can be coded using numbers, these numbers cannot be ordered meaningfully).
- Examples: Biological Sex, Political parties.
Ordinal Scale:
- Description: Attributes can be ordered.
- Characteristics:
  - Has names (like nominal).
  - Values can be ordered.
  - The differences between values are not comparable; the distance between values is subjective.
- Example: Rank or order of winners (e.g., 1st, 2nd, 3rd place – the difference between 1st and 2nd might not be the same as between 2nd and 3rd in actual performance).
Interval Scale:
- Description: Distance is meaningful.
- Characteristics:
  - Has names (like nominal).
  - Values can be ordered (like ordinal).
  - Meaningful distance between values.
  - No true absolute zero point. A true absolute zero point means that the zero point on the measurement scale signifies the complete absence of the variable ( $0$ = nothing exists).
- Examples: Temperature (Celsius or Fahrenheit). For instance, $0^ ext{C}$ does not mean the absence of heat; it's just a point on the scale.
Ratio Scale:
- Description: Absolute zero; the strongest level.
- Characteristics:
  - Has names (like nominal).
  - Values can be ordered (like ordinal).
  - Meaningful distance between values (like interval).
  - Has an absolute zero point. This means that a value of zero genuinely indicates the absence of the quantity being measured.
- Examples: Weight, Number of correct answers (zero correct answers means an absence of correct answers).

Comparison of Levels of Measurement

Characteristic	Nominal	Ordinal	Interval	Ratio
Name?	$✔$	$✔$	$✔$	$✔$
Can be ordered?	$✘$	$✔$	$✔$	$✔$
Meaningful Distance?	$✘$	$✘$	$✔$	$✔$
Absolute Zero point?	$✘$	$✘$	$✘$	$✔$

Learning Check: Levels of Measurement

Question: A study assesses the optimal size (number of other members) for study groups. The variable "Size of group" is …
- A. Discrete and interval
- B. Continuous and ordinal
- C. Discrete and ratio
- D. Continuous and interval
Thought Process:
- Discrete vs. Continuous: Can group size take on any value (e.g., $2.5$ members)? No, it must be whole numbers, so it's discrete.
- Level of Measurement:
  - Names? Yes ( $1$ member, $2$ members, etc.).
  - Can be ordered? Yes (a group of $3$ is larger than $2$ ).
  - Meaningful distance? Yes (the difference between a $1$ and $2$ -person group is the same as between $2$ and $3$ ).
  - Absolute zero point? Yes, zero members means no group exists.
- Therefore, it is a ratio variable.
Answer: C. Discrete and ratio.

Algebra on Variables

Variables are typically indicated using capital letters (e.g., $X$ or $Y$ ).
Standard mathematical operations and notation can be applied to variables (e.g., $+, -, imes, rac{X}{Y}, X^2, ext{ and } rac{ ext{}}{X}$
Example with variables:
Given scores for $X$ and $Y$ :
| X | Y |
| - | - |
| 1 | 3 |
| 2 | 4 |
- $X+Y$
  - For first row: $1+3=4$
  - For second row: $2+4=6$
- $X+2$
  - For first row: $1+2=3$
  - For second row: $2+2=4$
- $XY$
  - For first row: $1 imes 3=3$
  - For second row: $2 imes 4=8$
- $2Y$
  - For first row: $2 imes 3=6$
  - For second row: $2 imes 4=8$

Summation Notation

Many statistical procedures involve summing (adding up) a set of scores.
The summation sign $ext{}$ stands for summation.
ext{X} = X1 + X2 + X3 + ext{ ext{ ext{ }}+ Xn
- The $ext{}$ is followed by a symbol or equation that defines what is to be summed.

Order of Operations in Statistics

Follow the standard order of operations, with summation having a specific placement:

Parentheses: Simplify expressions inside parentheses first.
Exponents: Perform operations involving exponents ( $X^2$ ) and roots ( $ext{}$ Next.
Multiplication and Division: Perform from left to right.
Summation ( $ext{}$ ): Summation operations are performed after operations within parentheses, squaring, and multiplication or division.
Addition and Subtraction: Perform from left to right. Summation is done before other separate addition or subtraction outside of summation.

Learning Check: Order of Operations

Question: $ext{(X^2 + 47)}$ instructs you to …
- A. Square each score and add $47$ to it, then sum those numbers.
- B. Square each score, add up the squared scores, then add $47$ to that sum.
- C. Add $47$ to each score, square the result, and sum those numbers.
- D. Add up the scores, square that sum, and add $47$ to it.
Explanation applying order of operations:
1. Look inside the parentheses first: $X^2 + 47$
2. Within the parentheses, perform the exponent first: $X^2$
3. Then perform the addition within the parentheses: $+ 47$
4. Finally, perform the summation (the $ext{}$ outside the parentheses):
Answer: A. Square each score and add $47$ to it, then sum those numbers.

Practice with Summation Notation

Given scores for $X$ and $Y$ :
| X | Y |
| - | - |
| 1 | 3 |
| 2 | 4 |

$ext{X}$ :
- Sum of all $X$ values: $1+2 = 3$
$ext{X}+1$
- Sum of all $X$ values first, then add $1$ : $ext{X}+1 = (1+2)+1 = 3+1 = 4$
$ext{(X+1)}$
- Add $1$ to each $X$ value first, then sum the results: $(1+1)+(2+1) = 2+3 = 5$
$ext{(Y-1)^2}$
- For each $Y$ value: subtract $1$ , then square the result.
- For $Y=3$ : $(3-1)^2 = 2^2 = 4$
- For $Y=4$ : $(4-1)^2 = 3^2 = 9$
- Then sum these squared results: $4+9 = 13$
$ext{XY}$
- For each row: multiply $X$ by $Y$ .
- For first row: $1 imes 3 = 3$
- For second row: $2 imes 4 = 8$
- Then sum these products: $3+8 = 11$