Unit 1.10 – The Normal Distribution
The Normal Distribution:
The "bell curve" — a symmetric, unimodal, continuous probability distribution that’s fundamental in statistics.
The "bell curve" — a symmetric, unimodal, continuous probability distribution that’s fundamental in statistics.
🔍 What Is the Normal Distribution?
- Symmetric, single-peaked, bell-shaped curve
- Describes many real phenomena: heights, weights, SAT scores
- Mean, median, and mode are all equal at the center
- Completely described by its mean (\(\mu\)) and standard deviation (\(\sigma\))
📈 Visual Features of the Normal Curve
- Center: \( \mu \)
- Spread: \( \sigma \)
- Inflection points (change in curvature) at \( \mu \pm \sigma \)
- Tails approach but never touch the x-axis
The Normal Curve
\[
N(\mu, \sigma)
\]
Standard Normal: \( \mu = 0,\, \sigma = 1 \) (\( N(0,1) \))
Probability Density Function:
\[
f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{- \frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2}
\]
⚡ 68–95–99.7 (Empirical) Rule
- \( \approx 68\% \) of data falls within 1 SD (\( \mu \pm \sigma \))
- \( \approx 95\% \) within 2 SDs (\( \mu \pm 2\sigma \))
- \( \approx 99.7\% \) within 3 SDs (\( \mu \pm 3\sigma \))
- This is called the Empirical Rule
| Interval | % of Area |
|---|---|
| \(\mu \pm \sigma\) | 68% |
| \(\mu \pm 2\sigma\) | 95% |
| \(\mu \pm 3\sigma\) | 99.7% |
🧮 Working with Z-Scores
- A z-score standardizes a value by telling how many SDs above or below the mean a value is.
- Positive z = above mean, negative z = below mean
- Z-scores allow comparison across different normal distributions
Z-Score Formula
\[
z = \frac{x - \mu}{\sigma}
\]
Interpreting Z-Scores
- \( z = 0 \): at the mean
- \( z = 1 \): 1 SD above mean
- \( z = -2 \): 2 SD below mean, etc.
📐 Normal Distribution Calculation Steps
- Draw curve, mark \( \mu \), \( \sigma \), labeled axes
- Convert \( x \) to z-score: \( z = \frac{x - \mu}{\sigma} \)
- Find area/probability using z-table or calculator
- Shade the area representing the probability
💡 Tricks, Tips & Facts
- When asked to "estimate" proportions from a normal, use the 68–95–99.7 rule
- Negative z-scores indicate values below the mean
- If z is larger than 3 or less than -3, value is very unusual!
- Area under the normal curve equals 1 (or 100%)
- For AP Stats: Always sketch and label your curve; show z-score calculation
- Z-table values give cumulative area to the LEFT of z
❌ Common Mistakes
- Confusing area with actual data value (area = probability, not height)
- Misreading z-table (are you getting area left or right?)
- Forgetting to standardize (use z) before using z-table
- Assuming all bell shapes are normal (skewed or bimodal aren't!)
- Forgetting that probabilities are always between 0 and 1
Summary:
Unit 1.10 is all about the normal distribution—the foundation for inference in statistics. Know how to recognize, standardize, calculate, and interpret values and areas, and always relate back to context. The 68–95–99.7 rule, z-scores, and clear labeled sketches are must-knows for AP Stats!
Unit 1.10 is all about the normal distribution—the foundation for inference in statistics. Know how to recognize, standardize, calculate, and interpret values and areas, and always relate back to context. The 68–95–99.7 rule, z-scores, and clear labeled sketches are must-knows for AP Stats!