Introduction to Probability and Statistics

Understanding Variables and Data

The characteristics we measure within a population are called variables. A variable can take on different values for different elements, such as age, gender, or test scores.

Types of Variables

Variables are categorized based on the nature of the information they provide:

1. Qualitative (Categorical) Variables: These represent names or labels. They are further divided into:
   • Nominal: Categories with no inherent mathematical order (e.g., gender or eye color).
   • Ordinal: Categories that have a logical order or ranking (e.g., satisfaction levels like "low," "medium," or "high").
2. Quantitative (Numerical) Variables: These represent measurable quantities and are divided into:
   • Discrete: Countable values that are usually integers (e.g., the number of students in a class).
   • Continuous: Values that can take any possible number within a range, often involving measurements (e.g., height, weight, or temperature).

Sources of Data

The actual observed values of these variables are called data. Data can be classified by how it was obtained:

• Primary Data: Data collected firsthand by the researcher specifically for the current study.
• Secondary Data: Pre-existing data that was collected by someone else for a different purpose.

Sampling Methods: Probability vs. Non-Probability

How we select our sample determines whether our results can be generalized to the whole population.

Probability (Random) Sampling

In probability sampling, every element in the population has a known, non-zero chance of being selected. This reduces bias and allows for the calculation of sampling error.

• Simple Random Sampling (SRS): Every individual is chosen entirely by chance, such as through a lottery or a random number generator.

  

• Systematic Sampling: Selecting every $k$-th element from a list after a random starting point.
  • Example: To pick 12 students from a class of 120, calculate the interval $k = 120 / 12 = 10$. If you randomly start at student #4, your sample will be students 4, 14, 24, and so on.
• Stratified Sampling: The population is divided into subgroups (strata) based on shared characteristics, and then a random sample is taken from each subgroup. This ensures all important groups are represented.
• Cluster Sampling: The population is divided into existing groups (clusters), such as classrooms. A few clusters are randomly selected, and every member within those chosen clusters is studied.

  

Non-Probability Sampling

In these methods, not every element has a chance of being selected. These are often used for pilot studies or when time and budget are limited, but they carry a higher risk of bias.

• Convenience Sampling: Selecting whoever is easiest to reach (e.g., asking people in a hallway).
• Purposive (Judgmental) Sampling: The researcher uses their expertise to select specific individuals who fit the study's purpose.
• Quota Sampling: Similar to stratified sampling, but the selection within groups is not random; researchers simply fill a "quota" for each category.
• Snowball Sampling: Used for hard-to-reach populations; existing participants refer the researcher to other potential participants.

Exam Tip: If a question asks whether a study can be "generalized" to a whole population, look at the sampling method. Probability sampling allows for generalization, while non-probability sampling generally does not.