AP Statistics · Unit 2 · Topic 2.11

The Normal Distribution Study Lab

A normal distribution is the symmetric, bell-shaped model described by its mean μ and standard deviation σ. Probabilities are areas under the curve. Pick a situation, find the area for a range of values, or work backward from a percentile to a value, with z-scores and calculator syntax for every step.

1. Areas and values

Choose what you know. "Find a probability" goes from values to an area. "Find a value" goes from an area (a percentile) back to a value.

z = (x − μ) ÷ σ x = μ + z · σ Total area = 1

Situations

Show the 68–95–99.7 rule

Mean μ

Standard deviation σ

Value a

Value b

Normal curve

Area (probability)

z-score(s)

Percentile

Unshaded area

Standardize first

Calculator syntax

2. Key ideas

Shape

Symmetric and bell-shaped

A normal curve is single-peaked and symmetric about μ, so the mean and median are equal. σ sets how spread out it is; the inflection points are one σ on each side of μ.

Empirical rule

68–95–99.7

In any normal distribution, about 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. Use it for quick estimates; use technology for exact areas.

z-scores

How many SDs from the mean

z = (x − μ)/σ. A positive z is above the mean and a negative z is below. Standardizing turns every normal distribution into the standard normal, N(0, 1).

Area = probability

Values to area

The proportion of values in an interval is the area under the curve over that interval. Since it is continuous, P(X < a) = P(X ≤ a).

Inverse

Area to value

To find a percentile, start from the area to the left, find the matching z with invNorm, then unstandardize: x = μ + zσ.

It is a model

Check the shape first

Only use normal calculations when the problem says the distribution is approximately normal or the data's graph supports it. Strongly skewed data do not follow these areas.

3. Calculator commands

Examples use X ~ N(100, 15).

Question	Area or value?	Command	Answer
P(X < 120)	Area to the left	normalcdf(−1E99, 120, 100, 15)	0.9088
P(X > 120)	Area to the right	normalcdf(120, 1E99, 100, 15)	0.0912
P(85 < X < 115)	Area between	normalcdf(85, 115, 100, 15)	0.6827
90th percentile	Value with 0.90 to its left	invNorm(0.90, 100, 15)	119.22
Top 5% cutoff	Value with 0.05 to its right	invNorm(0.95, 100, 15) (or area 0.05, RIGHT)	124.67
Middle 50%	Q₁ and Q₃	invNorm(0.25, 100, 15), invNorm(0.75, 100, 15)	89.88 to 110.12

4. On the FRQ

How to earn the points

Name the model with its parameters: "Let X = battery life; X is approximately N(μ = 40, σ = 5) hours."
Write the probability statement and show the z-score: "P(X > 48) = P(Z > (48 − 40)/5) = P(Z > 1.6) = 0.0548."
If you use calculator syntax, label the inputs: "normalcdf(lower = 48, upper = 1000, μ = 40, σ = 5)." Unlabeled syntax may not earn full credit.
Sketch the curve with the mean, the boundary value, and the shaded region. It catches left/right mistakes.
For a percentile, give the value in context: "A battery must last at least 46.4 hours to be in the top 10%."

Common mistakes

Using the area to the left when the question asks "more than" or "at least."
Plugging the area into invNorm as a right-tail area without telling the calculator, or giving a z-score when a value was asked for.
Using σ = variance. If you are given σ² = 25, then σ = 5.
Applying normal areas to a distribution that is clearly skewed.
Rounding z to one decimal early. Keep at least two decimals, or let technology do the whole calculation.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 2.11. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.