Module 4: Continuous Probability Distributions
Master the normal and uniform distributions
Prerequisites: Modules 2 & 3
This module builds on concepts from Module 2: Descriptive Statistics (mean, standard deviation) and Module 3: Probability Basics (probability distributions, expected value). Make sure you're comfortable with these topics before starting.
Learning Objectives
By the end of this module, you will be able to:
- Recognize and describe the characteristics of a normal distribution
- Apply the Empirical Rule (68-95-99.7 rule) to normal distributions
- Calculate and interpret z-scores
- Use the standard normal distribution table (z-table)
- Find probabilities for normal distributions
- Find values corresponding to given percentiles
- Determine when normal approximation is appropriate
- Understand the uniform distribution and calculate probabilities
- Compare and contrast normal and uniform distributions
- Apply continuous probability distributions to real-world problems
Lessons
Introduction to the Normal Distribution
Learn about the bell curve, its properties, and why it's so important in statistics.
- What makes a distribution "normal"
- Parameters: mean (μ) and standard deviation (σ)
- The Empirical Rule (68-95-99.7 rule)
- Real-world examples of normal distributions
Standard Normal Distribution & z-scores
Master the standard normal distribution and learn to calculate z-scores.
- The standard normal distribution (μ = 0, σ = 1)
- z-score formula: z = (x - μ) / σ
- Interpreting z-scores
- Using the z-table
Finding Probabilities with the Normal Distribution
Learn to find areas under the normal curve and calculate probabilities.
- Finding P(X < a) and P(X > a)
- Finding P(a < X < b)
- Working with z-tables
- Finding percentiles and critical values
Normal Approximation & Applications
Apply the normal distribution to real-world problems and learn when to use it.
- Checking for normality (normal probability plots, histograms)
- Normal approximation to the binomial
- Continuity correction
- Real-world applications (test scores, heights, measurements)
The Uniform Distribution
Learn about the uniform distribution and when all outcomes are equally likely.
- Properties of the uniform distribution U(a, b)
- Calculating probabilities with uniform distributions
- Mean, variance, and standard deviation formulas
- Comparing uniform and normal distributions
Assessments & Practice
Test your understanding and practice what you've learned.
Study Tips for Module 4
- Visual learning: Draw bell curves for every problem to visualize the area you're finding
- Master z-scores first: Everything in this module builds on understanding z-scores
- Practice with z-tables: Get comfortable reading both left-tail and right-tail probabilities
- Memorize the Empirical Rule: 68-95-99.7 is incredibly useful for quick estimates
- Check your work: Probabilities must be between 0 and 1; z-scores tell you how many SDs from mean