Overview
This lecture explains the normal distribution (bell curve), details the 68-95-99.7 rule, and demonstrates solving probability problems using both percentages and calculus-based integration.
The Normal Distribution Curve
- The normal distribution is a symmetric, bell-shaped curve centered at the population mean (μ).
- Standard deviation (σ) measures the spread; points at μ ± nσ mark intervals.
- The total area under the curve is 1 (100% probability).
The 68-95-99.7 Rule (Empirical Rule)
- 68.27% of data falls within 1 standard deviation of the mean (μ ± 1σ).
- 95.45% of data falls within 2 standard deviations (μ ± 2σ).
- 99.73% of data falls within 3 standard deviations (μ ± 3σ).
- Each side between mean and 1σ: 34.13%; between 1σ and 2σ: 13.59%; between 2σ and 3σ: 2.14%; beyond 3σ: 0.135%.
Using Areas for Probability
- Probability of an interval is the area under the curve for that interval.
- The curve is symmetric; left and right regions are equal.
- Probability between A and B: add up the areas for those regions.
Example Problems
- For test scores with mean = 75, σ = 7, use intervals to find probabilities:
- Between 68 (μ - 1σ) and 82 (μ + 1σ): 68.27% of students.
- Between 61 and 89 (μ ± 2σ): 95.45% of students (~764 out of 800).
- Between 54 and 75: Add areas for each region (about 49.87%).
- Greater than or equal to 96 (μ + 3σ): 0.135% (~1 student).
Calculus Approach
- The exact probability is the definite integral:
[
P(A < X < B) = \int_{A}^{B} \frac{e^{-\frac{(x-\mu)^2}{2\sigma^2}}}{\sigma \sqrt{2\pi}} dx
]
- Change limits (A, B), plug in μ and σ; use calculators (e.g., Wolfram Alpha).
Using Online Calculators
- Online integration calculators (e.g., Wolfram Alpha) simplify finding probabilities for given intervals.
- Only change integration limits; keep formula and parameters (μ, σ) the same.
Key Terms & Definitions
- Normal Distribution — Symmetrical probability distribution shaped like a bell curve.
- Mean (μ) — The average value of the distribution.
- Standard Deviation (σ) — Measures the dispersion from the mean.
- Empirical Rule — The 68-95-99.7% rule describing data spread.
- Probability — Likelihood of an event, represented by area under the curve.
- Definite Integral — Calculus method to find exact area/probability between two points.
Action Items / Next Steps
- Memorize the 68-95-99.7 rule percentages.
- Practice calculating probabilities using both the area method and definite integrals.
- Try sample problems using an online calculator (e.g., Wolfram Alpha) for definite integrals with different μ, σ, and limits.