Normal Distribution – AP Statistics

Learning Objectives

The Bell Curve: Measured traits (height, test scores) cluster perfectly symmetrically around the mean ($\mu$).

Variance & Spread ($s$): The Standard Deviation ($\sigma$) dictates how wide the bell is. A small $\sigma$ means tightly clustered data.

The Empirical Rule: $68\%$ of data falls within $\pm 1\sigma$. $95\%$ within $\pm 2\sigma$. $99.7\%$ within $\pm 3\sigma$. Any point beyond $\pm 3\sigma$ is a severe outlier!

Tags

NormalStatisticsZ-ScoreStandard Deviation

       PROBABILITY DENSITY FUNCTION $N(\mu, \sigma)$

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

Distribution Parameters

Mean ($\mu$) = 100

Standard Dev ($\sigma$) = 15

Z-Score Calculator & Shading

Target X = 115

$Z = \frac{X - \mu}{\sigma}$

$X$ minus mean: 15.0

Z-Score Value: 1.00

CUMULATIVE PROBABILITY $P(x \leq X)$

Percentile Area: 84.1%

The Z-Score: A Z-score tells you EXACTLY how many standard deviations away from the mean you are. A score of $Z=1.0$ (like an IQ of 115) puts you above 84.1% of the population!