Normal Distribution for AP Statistics Explained | 2025 Exam Guide

Introduction: Why Normal Distribution Is Essential

The normal distribution is one of the most important concepts in AP Statistics. It’s the backbone for probability, inference, and z-scores — and it appears every year on the AP exam.

This guide will teach you to:

• Recognize normal distributions.
• Use the 68-95-99.7 rule.
• Calculate z-scores.
• Use your calculator for probabilities.
• Apply normal models on multiple-choice and FRQ problems.
• Practice with RevisionDojo’s resources.

Step 1: What Is the Normal Distribution?

• A bell-shaped curve that is symmetric around the mean.
• Defined by two parameters:
  • μ (mean): center.
  • σ (standard deviation): spread.
• Total area under curve = 1 (represents all possible outcomes).

👉 On AP Stats: Normal distribution is used for modeling real-world data and approximating probabilities.

Step 2: The Empirical Rule (68-95-99.7)

For any normal distribution:

• About 68% of data falls within 1 standard deviation (μ ± 1σ).
• About 95% within 2σ.
• About 99.7% within 3σ.

Example: If μ = 100 and σ = 15:

• 68% of values between 85 and 115.
• 95% between 70 and 130.
• 99.7% between 55 and 145.

👉 RevisionDojo’s Visual Normal Curve Tool helps you master curve shading.

Step 3: Z-Scores

Formula:
z = (x – μ) / σ

• Measures how many standard deviations a value is from the mean.
• Positive z = above mean.
• Negative z = below mean.

Example: SAT score 1300, μ = 1050, σ = 150 → z = (1300–1050)/150 = 1.67.

👉 Use z-scores to compare across different distributions.

Step 4: Calculator Functions

TI-84/TI-Nspire Commands:

• normalcdf(lower, upper, μ, σ) → Probability between bounds.
• invNorm(area, μ, σ) → Value corresponding to percentile.

Example: P(x > 120) with μ = 100, σ = 10:

• normalcdf(120, 1E99, 100, 10) → ~0.0228.

👉 RevisionDojo’s calculator guides show keystrokes step by step.

Step 5: Common Exam Applications

1. Finding probabilities from raw scores.
2. Finding percentiles.
3. Comparing z-scores across different data sets.
4. Checking normality before using inference procedures.

Step 6: Checking for Normality

On FRQs, you may need to justify if data is “approximately normal.”

• Look at histograms, boxplots, or normal probability plots.
• Symmetry and lack of skew/outliers = normal enough.

👉 Even if not perfect, large samples let you apply the Central Limit Theorem.

Step 7: Practice Question Walkthroughs

Question 1

Heights of students are normally distributed with μ = 65 in, σ = 3 in. What proportion are taller than 70 in?

Solution:
z = (70–65)/3 = 1.67 → P(z > 1.67) ≈ 0.0475.

Answer: ~4.8%.

Question 2

SAT Math scores are N(500, 100). What score corresponds to the 90th percentile?

Solution:
invNorm(0.90, 500, 100) ≈ 628.

Answer: 628.

Question 3

If μ = 60, σ = 8, what’s probability between 52 and 68?

Solution:
z-scores: (52–60)/8 = –1, (68–60)/8 = 1. → Between z = –1 and 1 = 68%.

Answer: 68%.

Step 8: Common Mistakes to Avoid

• Forgetting to standardize values with z.
• Mixing up percentile with probability.
• Forgetting to use bounds (normalcdf needs lower and upper).
• Misinterpreting calculator outputs (round carefully).

👉 RevisionDojo’s Mistake Tracker highlights these pitfalls.

Step 9: Connecting Normal Distribution to Inference

Why normal matters:

• Sampling distributions approximate normal with large n.
• Confidence intervals and hypothesis tests rely on z/t distributions.
• Normal models justify using inference procedures.

Step 10: Study Strategy for Mastery

• Draw curves when solving — visual memory helps.
• Practice z-score conversions daily.
• Use calculator drills until commands are second nature.
• Connect normal distributions to probability + inference.

RevisionDojo Resources

• Visual Normal Curve Tool: Curve shading and rule practice.
• Calculator Walkthroughs: For normalcdf and invNorm.
• Mistake Tracker: Logs common student errors.
• Practice Bank: AP-style problems with worked solutions.

👉 Check out RevisionDojo’s Normal Distribution Hub here.

Frequently Asked Questions (FAQs)

Q: Is the normal distribution always required in AP Stats?
A: No, but it underlies many inference methods.

Q: Do I need to memorize the 68-95-99.7 rule?
A: Yes — it appears frequently in MCQs and FRQs.

Q: Can I use normal distribution for small samples?
A: Only if data is approximately normal. Otherwise, use t-distribution.

Q: What calculator commands should I know?
A: normalcdf for probabilities, invNorm for percentiles.

Q: How do I check normality on the exam?
A: Use plots — symmetry, bell-shape, no major outliers.

Final Thoughts

Normal distribution is a core building block in AP Statistics. It appears in probability, sampling distributions, and inference.

Remember:

• Use z-scores to standardize.
• Apply the 68-95-99.7 rule for quick approximations.
• Practice calculator functions until automatic.
• Always interpret in context.

With RevisionDojo’s curve tools, calculator guides, and problem banks, you’ll turn normal distribution from a confusing formula into a powerful exam weapon.