Normal distributions and the empirical rule

What the exam asks

• Identify parameters from a curve: Locate the mean $\mu$ at the center of symmetry and the standard deviation $\sigma$ at the inflection points of a given normal curve.
• Compare normal curves: Explain how changing $\mu$ shifts a curve left or right and how changing $\sigma$ makes the curve taller and narrower or shorter and wider.
• Apply the empirical rule: Calculate approximate percentages or probabilities for intervals bounded by $1$, $2$, or $3$ standard deviations from the mean without needing a calculator or table.
• Standardize values: Relate any normal distribution to the standard normal distribution ($\mu = 0, \sigma = 1$) using $z$-scores.

Core ideas

Continuous random variables and normal curves

A continuous random variable can take on any numerical value in an interval on a number line (such as height, time, or weight). Unlike discrete random variables, the probability of a continuous random variable equaling one exact value is $0$; instead, probabilities correspond to areas under a density curve over an interval, where the total area under the curve is always $1$ (or $100\%$). Consequently, $P(X < c)$ is the same as .

A normal distribution is modeled by a continuous density curve that is unimodal, roughly bell-shaped, and symmetric about its mean.

Parameters $\mu$ and $\sigma$

Every normal distribution is completely specified by two parameters: its mean $\mu$ and its standard deviation $\sigma$.

• Mean ($\mu$): Located at the exact center of symmetry (the peak of the curve). Because of symmetry, the mean equals the median. Changing $\mu$ shifts the entire curve horizontally along the axis without changing its shape.
• Standard deviation ($\sigma$): Measures variability and equals the horizontal distance from $\mu$ to the two inflection points—the points on either side of the peak where the curve changes from curving downward (steepening) to curving upward (flattening out).