Normal Distribution – AP Statistics

Learning Objectives

Describe normal distribution by its mean (μ) and standard deviation (σ)

Apply the 68-95-99.7 (Empirical) Rule

Calculate z-scores and find probabilities

Understand how μ and σ affect the shape of the curve

Key Equations

\( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

\( z = \frac{x - \mu}{\sigma} \)

\( P(\mu-\sigma < X < \mu+\sigma) \approx 68\% \)

\( P(\mu-2\sigma < X < \mu+2\sigma) \approx 95\% \)

\( P(\mu-3\sigma < X < \mu+3\sigma) \approx 99.7\% \)

Why It Matters

The normal distribution is the most important probability distribution in statistics. It models everything from test scores to measurement errors. The Central Limit Theorem guarantees that sample means follow it, making it the backbone of inference.

Tags

normal distributionbell curvez-scoreempirical rulestandard deviationAP Statistics

μ0

σ1

P(a < X < b)—

z(a)—

z(b)—

Mean (μ) 0

Std Dev (σ) 1.0

Shade from (a) -1.0

Shade to (b) 1.0

Quick Shade

Show

Probability

P = —

Quick Quiz

For N(100, 15), approximately what percentage of data falls between 70 and 130?