Unit 2 · Topic 2.11

2.11 The Normal Distribution

The normal distribution is the bell-shaped model used all through inference. It's set by its mean μ and standard deviation σ. You'll use the empirical rule for quick estimates, then z-scores with a table or technology to find any area or any percentile.

Key terms

normal distribution
standard normal (μ = 0, σ = 1)
empirical rule (68–95–99.7)
area under the curve
percentile

Continuous random variables and the normal curve

A continuous random variable can take any value in an interval, like a person's height. Probabilities are areas under a density curve, and the total area is 1. The probability of any single exact value is 0, so P(X < 180) and P(X ≤ 180) are the same.

A normal distribution is continuous, unimodal, symmetric and bell-shaped. Its center is μ, and σ controls the spread: a small σ gives a tall, narrow curve; a large σ gives a short, wide one. Many real measurements, like heights or test scores, are close to normal. The standard normal distribution has μ = 0 and σ = 1.

The empirical rule

For any normal distribution, about 68% of values are within 1σ of the mean, about 95% within 2σ, and about 99.7% within 3σ. This is the 68–95–99.7 rule.

If heights are normal with μ = 170 cm and σ = 8 cm, about 95% are between 154 and 186 cm. Because the curve is symmetric, about 2.5% are above 186 cm and 2.5% are below 154.

Only use the empirical rule when the distribution is approximately normal. For skewed data the percentages are wrong.

Finding areas

To find P(X < x), standardize with z = (x − μ)/σ and look up z in Table A (it gives the area to the left), or use normalcdf on a calculator or in Desmos. For an area to the right, subtract from 1. For an area between two values, subtract the two left areas.

Always sketch the curve and shade the area you want. It catches most sign and direction mistakes.

Finding percentiles and cutoffs

To go from an area to a value, work backward. Find the z with the given area to its left (from the inside of Table A, or invNorm), then x = μ + zσ.

Percentiles also let you compare positions across different normal distributions. A score at the 84th percentile on one test and the 70th on another tells you directly which is relatively better, just like comparing z-scores.

Match the inequality to the area:

Lowest p%: find x so the area to the left of x is p%.
Highest p%: the area to the left of x is (100 − p)%.
Middle p%: split the leftover area equally, (100 − p)/2 % in each tail.
Most extreme p% on both sides: put p/2 % in each tail.

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Area to the left and between
Heights of adult women in a region are approximately normal with μ = 170 cm and σ = 8 cm. Find the proportion shorter than 180 cm and the proportion between 160 and 180 cm.
Show the solutionHide the solution
1. Step 1: z for 180: (180 − 170)/8 = 1.25. Table A: area left of 1.25 is 0.8944. So P(X < 180) ≈ 0.894.
2. Step 2: z for 160: (160 − 170)/8 = −1.25. Area left is 0.1056.
3. Step 3: P(160 < X < 180) = 0.8944 − 0.1056 = 0.7888 (technology: 0.7887).
Answer: About 89.4% are shorter than 180 cm; about 78.9% are between 160 and 180 cm.
Example 2Calculator allowed
Find a percentile
Using the same distribution, how tall must a woman be to be in the tallest 10%?
Show the solutionHide the solution
1. Step 1: The tallest 10% means 90% of the area is to the left of the cutoff.
2. Step 2: Find z with area 0.90 to the left: z ≈ 1.28 (invNorm(0.90) = 1.2816).
3. Step 3: x = 170 + 1.2816 × 8 ≈ 180.3 cm.
Answer: About 180.3 cm or taller.
Example 3Calculator allowed
Trap: right-tail area
Using the same distribution, a student finds the proportion of women taller than 185 cm by computing z = 1.875 and reporting 0.970. What's the correct answer?
Show the solutionHide the solution
1. Step 1: Table A and normalcdf from the left give P(X < 185) ≈ 0.970.
2. Step 2: "Taller than" is the right tail, so subtract from 1.
3. Step 3: P(X > 185) ≈ 1 − 0.970 = 0.030.
Answer: About 0.030. The student gave the area to the left instead of the right.

Common mistakes

Using the area to the left when the question asks for "greater than."
Using the empirical rule or normal calculations for a distribution that isn't approximately normal.
Flipping the sign of z, or using the variance instead of σ when standardizing.
Writing only calculator syntax. Show the distribution, the boundary and the direction.

On the exam

On free response, communicate clearly: "X is approximately normal with μ = 170 and σ = 8; P(X > 185) = P(z > 1.875) ≈ 0.030." Include a labeled sketch if it helps.
Percentile questions are common in multiple choice: decide first whether the area you need is to the left, right or in the middle.

Connected topics

Videos

AP Stats 2.B.3 - The Empirical Rule & Standard Normal Curve
Skew The ScriptWatch on YouTube (opens in a new tab)
The NORMAL Distribution EXPLAINED [AP Statistics Topic 1.10]
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)
The Normal Distribution and the 68-95-99.7 Rule (5.2)
Simple Learning ProWatch on YouTube (opens in a new tab)
AP Stats 2.B.4 - Normal Calculations
Skew The ScriptWatch on YouTube (opens in a new tab)
The Normal Distribution: Crash Course Statistics #19
CrashCourseWatch on YouTube (opens in a new tab)
AP Statistics: Normal Distribution Made EASY (Desmos, TI-84 & NumWorks Tutorial)
Michael Porinchak - AP Statistics & AP PrecalculusWatch on YouTube (opens in a new tab)