Methods of Categorizing Variables in Statistics

The Importance of Categorizing Variables

Categorizing variables is foundational to the study of statistics because it determines the mathematical operations and analytical techniques that can be applied to data.
Researchers cannot use every statistical tool with every kind of variable; each method is limited to specific categories of data.
Example: Measuring the favorite color of a sample of $50$ people allows for different statistical possibilities compared to measuring the height of a sample of $50$ people.
There are two primary ways to categorize variables: a simple binary classification (Discrete vs. Continuous) and a more nuanced classification (The Four Scales of Measurement).

Discrete vs. Continuous Variables

Discrete Variables * Definition: A discrete variable consists of a limited number of categories. Significantly, no intermediate value is possible between any two adjacent categories. * The "Jump" Characteristic: Discrete variables move from one score to the next in sudden jumps rather than gradual transitions. * Example: Number of Siblings: You can have $2$ or $3$ siblings, but it is impossible to have $2.1$ or $2.463$ siblings. You remain at $2$ siblings until the exact moment a third is born. * Example: College Units: A student might have completed $60$ units and then jump to $61$ or $72$ units at the end of a semester. A counselor would find a value like $60.532$ units impossible. * Example: Recipe Ingredients (Counted): A recipe calling for eggs is discrete. You add $4$ or $5$ eggs, not $4.732$ eggs. * Identification Tip: Discrete variables are typically things you count (e.g., "number of…").
Continuous Variables * Definition: Continuous variables have an infinite number of categories. Between any two categories, there is an infinite number of possible intermediate values. * The "Flow" Characteristic: Continuous variables do not jump; they pass through every decimal place (e.g., $2.1$ , $2.2$ , $2.01$ , $2.001$ ) as they increase or decrease. * Example: Sugar in a Recipe: Unlike eggs, sugar is measured. A recipe might call for $2$ or $3$ cups, but one could technically measure out $2.463$ cups with a precise enough tool. * Example: Height and Weight: These are naturally continuous. To grow from $70$ inches to $71$ inches, a person must pass through every trillionth of an inch in between. * Identification Tip: Continuous variables are typically things you measure rather than count. * Philosophical Connection: The concept of passing through an infinite number of subdivisions is related to Zeno's Paradox (attributed to the ancient Greek philosopher Zeno or Xeno).
Gray Areas and Nuance (The Case of Age vs. Time) * Time: Time is fundamentally continuous. It passes through infinite fractions of a second (e.g., $10.463$ seconds). * Age: While age measures the continuous passage of time, it is culturally treated as a discrete variable. People state they are $27$ years old until the exact day or minute they become $28$ . We generally do not use values like $27.529$ years old in standard conversation.

The Four Scales of Measurement (NOIR)

The four scales—Nominal, Ordinal, Interval, and Ratio—provide a gradation of detail. They follow the mnemonic NOIR. This hierarchy is cumulative: each higher scale possesses all the properties of the levels below it plus one unique characteristic.
Nominal Scale * Definition: Sets of categories that serve as names only, with no particular logical order or structure. * Etymology: "Nom" means name. * Example: Religion: In a survey with $10$ checkboxes for religion (e.g., Protestant, Catholic, Buddhist, Muslim, None), there is no "correct" mathematical order to list them. * Example: Academic Majors: Psychology, Sociology, and Nursing are just names given to different sets of classes. * Other Examples: Gender, ethnicity, and nationality.
Ordinal Scale * Definition: Sets of categories that possess a built-in, logical order. * Cumulative Property: Nominal properties (names) + Order. * Example: Military Ranks: Ranks like Private, Colonel, and General have a clear hierarchy that must be respected. * Example: Year in School: Freshman, Sophomore, Junior, and Senior follow a fixed sequence. * Example: Sizes: Small, Medium, Large. * Example: Olympic Medals: Gold ( $1^{st}$ ), Silver ( $2^{nd}$ ), and Bronze ( $3^{rd}$ ). * Key Limitation: The distance (interval) between categories is not necessarily equal. In a race, the time difference between Gold and Silver might be $10$ seconds, while the difference between Silver and Bronze might be only $0.5$ seconds.
Interval Scale * Definition: Ordered categories where the distance (interval) between any two adjacent categories is consistent across the entire scale. * Cumulative Property: Ordinal properties (order) + Consistent Intervals. * Example: Temperature (Fahrenheit and Celsius): The difference between $80^{\circ}$ and $81^{\circ}$ represents the same amount of heat change as the difference between $250^{\circ}$ and $251^{\circ}$ . A degree is always a degree, regardless of the starting point. * Example: Money (as a concept of interval): On a scale of earnings, the interval of $\$1$ ( $100\text{ cents}$ ) is the same whether you are moving from $\$10,000$ to $\$10,001$ or from $\$1,000,000$ to $\$1,000,001$ .
Ratio Scale * Definition: An interval scale with a meaningful, absolute zero point, representing the complete absence of the variable. * Cumulative Property: Interval properties (consistent intervals) + Absolute Zero. * Negative Values: A ratio scale cannot have negative numbers because you cannot have less than "nothing." * Example: Amount of Sugar: Zero cups of sugar means no sugar exists in the cup. You cannot have negative $2$ cups. * Example: Income from a Single Job: You can earn $\$0$ if you did not work, but you cannot earn negative money from that specific job. * Example: Temperature in Kelvins: Unlike Fahrenheit or Celsius (where zero is just another number and negatives are possible), $0\text{ Kelvin}$ is absolute zero. It represents the point where all molecular motion ceases (average kinetic energy is $0$ ). This corresponds to approximately $-253^{\circ}\text{ Celsius}$ .