Study Notes on Discrete and Continuous Variables
Discrete and Continuous Variables
Introduction to Variables
Variables in a study can be characterized by the types of values they can assume.
The type of values influences the statistical procedures for summarizing or making inferences.
Discrete Variables
Definition: Discrete variables consist of separate, indivisible categories.
Characteristics:
No intermediate values exist between two adjacent categories.
Example: Number of questions answered correctly on a 10-item multiple-choice quiz.
Between values 7 (seven correct) and 8 (eight correct), no intermediate values can be observed.
Common Characteristics:
Typically represented by whole, countable numbers (integers).
Examples include:
Number of children in a family.
Number of students attending a class (e.g., 18 students one day, 19 the next).
Cannot observe a value between 18 and 19 students.
Qualitative Observations:
Discrete variables can also include qualitative distinctions.
Examples:
Birth order (first-born vs. later-born).
Occupations (nurse, teacher, lawyer, etc.).
Academic major (art, biology, chemistry, etc.).
Continuous Variables
Definition: Continuous variables are not limited to a fixed set of separate categories.
Characteristics:
Continuous variables can be divided into an infinite number of fractional parts.
Example: Measurement of time (hours, minutes, seconds, fractions of seconds).
Detailed Explanation:
Continuous variables can represent an infinite number of values between any two observed values.
Example: Measuring weights in a diet study.
Visual representation: Continuous line with infinite possible points, no gaps between neighboring points.
For any two distinct points on a continuous variable line, a third value can always be found between them.
Weight Measurement Example:
A measurement of 150.5 does not strictly belong to 150 or 151 but resides at the boundary.
Placement depends on rounding rules:
Round up to 151 or round down to 150.
Considerations for Continuous Variables
Unique Measurements:
Identical measurements between different individuals should be rare, indicating true continuity.
If tied scores emerge, either:
The variable may not be truly continuous.
The measurement procedure might be imprecise (restricted to discrete values).
Measurement Categories:
Researchers must define measurement categories on the measurement scale.
Example: Weight measured to the nearest pound (categories: 149, 150, etc.).
Each measurement category is an interval defined by boundaries (real limits).
For a score of 150 pounds:
Lower real limit: 149.5
Upper real limit: 150.5
Individuals with weight between these limits (e.g., 149.6 and 150.3) are both assigned a score of 150.
Real Limits
Definition: Real limits are boundaries for intervals of scores on a continuous number line.
Characteristics:
Each score has two real limits:
Upper Real Limit: Top of the interval.
Lower Real Limit: Bottom of the interval.
Application:
Real limits apply to any measurement of continuous variables.
Example: Measuring time to the nearest tenth of a second.
Categories: 31.0, 31.1, 31.2, etc.
A score of X = 31.1 seconds corresponds to an interval defined by:
Lower real limit: 31.05
Upper real limit: 31.15
Real limits are essential for constructing graphs and performing calculations with continuous scales.
Important Distinction
Continuous vs. Discrete Measurement:
Terms ‘continuous’ and ‘discrete’ refer to the underlying variables and not the scores obtained from measurements.
Example: Height measured to the nearest inch produces discrete scores (60, 61, 62, etc.) but is fundamentally a continuous variable.
Key to determining variable type:
Continuous variables can be divided into any number of fractional parts (e.g., height can be measured to the nearest inch, 0.5 inch, or 0.1 inch).
Various measurement approaches:
Pass/fail system classifying students' knowledge is discrete.
A 10-point quiz: 11 categories (0 to 10) is discrete.
A 100-point exam: 101 categories (0 to 100) remains discrete.
Allowing choice in precision or number of categories indicates a continuous variable.
Conclusion
Understanding the differences between discrete and continuous variables is crucial for choosing appropriate statistical methods and interpreting study results.