Normal probabilities, percentiles and interval boundaries

What the exam asks

• Calculate the probability or proportion of values falling below, above, or between specific boundaries in a normal distribution.
• Determine the cutoff value(s) ($x_a$ or $x_b$) corresponding to a given percentile, tail area, or middle percentage.
• Compare relative standing of observations within the same distribution or across two different normal distributions using percentiles and proportions.
• Show complete distribution work on free-response questions by naming the distribution, stating its parameters ($μ$ and $σ$), specifying the boundary and inequality direction, and stating the numerical result.

Core ideas

Areas as probabilities

For a continuous random variable $X$ modeled by a normal distribution with mean $\mu$ and standard deviation $\sigma$, the probability that $X$ falls in an interval is the area under the normal curve over that interval.

Because a single vertical line has zero width and zero area, strict and non-strict inequalities have the exact same probability:

You can find normal probabilities in two ways:

1. Technology (fastest): Use your graphing calculator (normalcdf) or Bluebook’s built-in Desmos (normaldist(μ, σ).cdf(lower, upper)).
2. Standardizing ($z$-scores) and Table A: Convert boundary values using $z = \frac{x - \mu}{\sigma}$ and look up the cumulative left-tail area in Table A.

Finding interval boundaries (Inverse Normal)

When given a probability or percentage $p\%$ and asked for the boundary value(s) in the units of the problem, convert the target region into a cumulative left-tail area (or use a tail-aware calculator menu) to find the boundary $x$ directly, or find $z$ and solve: