Overview
This lecture covers discrete numerical data, emphasizing how to represent distributions, check their validity, calculate probabilities, graph them, and determine expected value (mean) for such variables.
Discrete Numerical Data & Similarity to Categorical Data
- Discrete numerical data consists of counting numbers as possible outcomes (e.g., 0, 1, 2, 3, 4).
- Both discrete numerical and categorical variables can list all possible outcomes.
Probability Distributions for Discrete Data
- A probability distribution table has outcomes in one row/column and their probabilities in another.
- Example: Number of books checked out (outcomes: 0, 1, 2, 3, 4) with corresponding probabilities.
Rules for Valid Probability Distributions
- Each probability must be between 0 and 1.
- The sum of all probabilities must equal 1.
- Only when both rules are satisfied can probabilities be found accurately using the table.
Calculating Probabilities from the Table
- To find the probability of a single outcome (e.g., checking out 0 books), use the table entry directly.
- For multiple outcomes (e.g., checking out 2 or more books), add together the probabilities of each relevant outcome.
- Rule of thumb: Add probabilities for multiple outcomes.
Graphing Discrete Probability Distributions
- The x-axis represents outcomes; the y-axis represents probabilities.
- Use line graphs or histograms, drawing a line from each outcome up to its probability.
- Ensure tick marks on the y-axis accommodate the highest probability value.
Expected Value (Mean) of Discrete Distributions
- Unlike a simple average, expected value weights each outcome by its probability.
- Formula: Multiply each outcome by its probability and sum the results (e.g., E(x) = 00.05 + 10.60 + 20.20 + 30.10 + 4*0.05).
- The expected value gives the typical or average outcome.
Key Terms & Definitions
- Discrete numerical data — Data with countable outcomes (e.g., 0, 1, 2...).
- Probability distribution — Table showing all outcomes and their probabilities.
- Expected value — Weighted average of all outcomes, using their probabilities.
Action Items / Next Steps
- Practice creating and checking probability distribution tables for validity.
- Calculate probabilities for single and multiple outcome events.
- Compute expected values for given discrete distributions.
- Try graphing a discrete probability distribution using line graphs.