Normal Distributions

The Normal Curve

A Normal distribution is a continuous probability distribution that is:

• Symmetric and bell-shaped
• Defined by two parameters: mean $\mu$ and standard deviation $\sigma$
• Notation: $X \sim N(\mu, \sigma)$

The Empirical Rule (68-95-99.7 Rule)

For any Normal distribution:

• 68% of data falls within $\mu \pm 1\sigma$
• 95% of data falls within $\mu \pm 2\sigma$
• 99.7% of data falls within $\mu \pm 3\sigma$

Z-Scores (Standard Normal)

A z-score tells how many standard deviations a value is from the mean:

$z = \frac{x - \mu}{\sigma}$

The Standard Normal distribution has $\mu = 0$ and $\sigma = 1$: $Z \sim N(0, 1)$

Interpretation: A z-score of 1.5 means the value is 1.5 standard deviations above the mean.

Finding Probabilities

To find $P(X < a)$ for $X \sim N(\mu, \sigma)$:

1. Standardize: $z = \frac{a - \mu}{\sigma}$
2. Use the Standard Normal table (Table A) or calculator
3. $P(X < a) = P(Z < z)$

Calculator: normalcdf(lower, upper, $\mu$, $\sigma$)

Finding Values from Probabilities (Inverse Normal)

Given a probability $p$, find the value $x$ such that $P(X < x) = p$:

1. Find $z^*$ from Table A (look up $p$ in the body)
2. Unstandardize: $x = \mu + z^* \cdot \sigma$

Calculator: invNorm($p$, $\mu$, $\sigma$)

Assessing Normality

Methods to check if data follows a Normal distribution:

1. Histogram/Dotplot: Should be roughly symmetric, bell-shaped
2. Normal Probability Plot (NPP): Points should fall approximately along a straight line
3. Empirical Rule check: ~68% within 1 SD, ~95% within 2 SD

AP Tip: Always state the distribution, show the z-score calculation, and sketch the curve with the area shaded when solving Normal distribution problems.