Mathfolis

The Normal Distribution

Unit 1 · Exploring One-Variable Data

What AP Stats asks here

The normal distribution is the workhorse of AP Statistics — symmetric, bell-shaped, parameterized by mean μ\mu and standard deviation σ\sigma. Two recurring AP questions: (1) what fraction of values lie in a stated range, answered via z-score then z-table or the empirical rule, and (2) given a percentile, find the cutoff value. The most common trap is applying the empirical rule to clearly non-normal (right-skewed) data.

Standardization

Z-score
z=x−μσz = \dfrac{x - \mu}{\sigma}
Inverse
x=μ+zσx = \mu + z\sigma

Empirical rule (68-95-99.7)

Within 1 SD
P(μ−σ≤X≤μ+σ)≈0.68P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.68
Within 2 SD
P(μ−2σ≤X≤μ+2σ)≈0.95P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.95
Within 3 SD
P(μ−3σ≤X≤μ+3σ)≈0.997P(\mu - 3\sigma \le X \le \mu + 3\sigma) \approx 0.997

Z-table convention

Table gives area to the left
P(Z≤z)P(Z \le z)
Right tail
P(Z>z)=1−P(Z≤z)P(Z > z) = 1 - P(Z \le z)
Between two z's
P(z1≤Z≤z2)=P(Z≤z2)−P(Z≤z1)P(z_1 \le Z \le z_2) = P(Z \le z_2) - P(Z \le z_1)
AP Tip: Z-scores are unit-free, so they let you compare performance across distributions with different scales — e.g., a 1.6-SD English score beats a 1.0-SD math score even when the raw English number is lower.
Caution: Before applying the empirical rule, check that the variable is approximately normal. Household income, time-to-event, and counts are right-skewed and will not obey 68-95-99.7.
Type 1

Empirical rule

Use 68-95-99.7 to find percentages between integer-SD boundaries. Remember the tail symmetry: 5% total beyond 2 SD splits into 2.5% per tail.

Example 1
Adult resting heart rates are approximately normal with μ=72\mu = 72 bpm and σ=8\sigma = 8 bpm. What percentage of adults have a resting heart rate below 56 bpm? (A) 2.5% (B) 5% (C) 16% (D) 32%

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Type 2

Z-score and percentile

Standardize the raw value to z, then use the standard-normal table for an area to the left (the percentile).

Example 2
SAT math scores are approximately normal with μ=528\mu = 528 and σ=117\sigma = 117. A student scores 700. Using a z-table (area left of z=1.47z = 1.47 is ≈0.9292\approx 0.9292), the student's approximate percentile is (A) 7th (B) 47th (C) 70th (D) 93rd

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Type 3

Inverse problem and cross-distribution comparison

Given a percentile, find the cutoff x=μ+zσx = \mu + z\sigma. To compare values from different normal distributions, standardize each to its own z-score.

Example 3
A student scores 85 on a math test (μ=75\mu = 75, σ=10\sigma = 10) and 78 on an English test (μ=70\mu = 70, σ=5\sigma = 5). Relative to other test-takers, on which test did the student perform better? (A) Math, because the raw score (85) is higher. (B) English, because zEng=1.6z_{\text{Eng}} = 1.6 exceeds zmath=1.0z_{\text{math}} = 1.0. (C) Both performances are equivalent because both z-scores are positive. (D) Cannot be determined without raw distributions.

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