The Normal Distribution

AP Statistics· 12 min read

1. Core Properties of the Normal Distribution★★☆☆☆⏱ 15 min

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Normal distributions are the most widely used probability model in statistics, as they naturally describe many real-world datasets like heights, test scores, and measurement errors. All normal distributions are symmetric, unimodal, and asymptotic, meaning the curve never touches the horizontal x-axis.

📘 Definition

Normal Distribution

$X \sim N (μ, σ^{2})$

A continuous probability distribution where the total area under the curve equals 1, and the mean, median, and mode are all equal at the center of the bell shape.

📐 Worked Example

Scores on a standardized exam follow a normal distribution with μ=500 and σ=100. Use the empirical rule to estimate the range that contains the middle 95% of all scores.

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Step 1: Recall the 95% interval for normal data falls 2 standard deviations above and below the mean.
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Step 2: Calculate lower bound: $μ - 2 σ = 500 - 2 * 100 = 300$
$300$
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Step 3: Calculate upper bound: $μ + 2 σ = 500 + 2 * 100 = 700$
$700$
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Final answer: The middle 95% of scores lie between 300 and 700.

✓ Quick check

Test your understanding of the empirical rule:

What percentage of normal data lies above μ + 2σ?
- 2.5%
- 5%
- 95%
- 16%
Reveal answer
2.5% —
Since 95% of data is within ±2σ, the remaining 5% is split equally between the two upper and lower tails.

2. Z-Scores and Standard Normal Standardization★★☆☆☆⏱ 12 min

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Z-scores let you convert any normal distribution to the standard normal N(0,1) distribution, so you can use shared probability tables or calculator functions for all normal problems instead of memorizing unique formulas for every possible mean and standard deviation.

📘 Definition

Z-Score

The number of standard deviations a raw observation $x$ is from the population mean.

z = \frac{x - \mu}{\sigma}

📐 Worked Example

A student scores 720 on the standardized exam from the previous example (μ=500, σ=100). Calculate their z-score.

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Step 1: Identify given values: $x = 720$ , $μ = 500$ , $σ = 100$
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Step 2: Substitute into z-score formula:
$z = \frac{720 - 500}{100} = 2.2$
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Final answer: The student's score is 2.2 standard deviations above the mean.

3. Normal CDF for Probability Calculations★★★☆☆⏱ 18 min

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The normal cumulative distribution function (CDF) returns the total area under the normal curve to the left of a specified x value, which equals the probability that a random observation is less than that x value. You can extend this to find interval or upper tail probabilities with simple arithmetic.

📐 Worked Example

Using the same exam distribution N(500, 100²), find the probability a random student scores between 600 and 700.

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Step 1: Recognize P(600 < X < 700) = P(X < 700) - P(X < 600)
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Step 2: Use normal CDF to find P(X < 700): that is the area left of 700, equal to 0.9772
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Step 3: Use normal CDF to find P(X < 600): that is the area left of 600, equal to 0.8413
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Step 4: Subtract to get the interval probability:
$0.9772 - 0.8413 = 0.1359$
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Final answer: ~13.6% of students score between 600 and 700.

4. Inverse Normal for Percentile Cutoffs★★★☆☆⏱ 15 min

📊 Graphing only

Inverse normal is the reverse operation of the normal CDF: you input a cumulative left area, mean, and standard deviation, and it returns the raw x value that corresponds to that percentile. This is used to find cutoff scores, margin of error bounds, and critical values for hypothesis tests later in the course.

📐 Worked Example

For the N(500, 100²) exam distribution, find the 90th percentile score, the value where 90% of all test takers score lower.

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Step 1: Confirm you are given cumulative left area = 0.9, μ=500, σ=100
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Step 2: Run inverse normal on your graphing calculator with these parameters
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Step 3: The output returns x ≈ 628.16
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Final answer: The 90th percentile exam score is approximately 628.

5. Common Pitfalls

Wrong move:

Confusing normal notation $N (μ, σ^{2})$ as mean followed by standard deviation

Why:

This leads you to use the full standard deviation value as the variance, producing z-scores that are half their correct size

Correct move:

Explicitly write out the mean and standard deviation separately before starting any calculation to avoid parameter mixups

Wrong move:

Applying the normal model to strongly skewed discrete data

Why:

Normal distributions are symmetric and continuous, so probability results will be wildly inaccurate for skewed datasets

Correct move:

Always confirm the data is approximately symmetric with no extreme outliers before using a normal model

Wrong move:

Forgetting to subtract left cumulative area from 1 for upper tail probabilities

Why:

Default normal CDF functions return left-side area, so unadjusted upper tail results will be greater than 0.5 and completely wrong

Correct move:

Label your target area as left, right, or interval before running any CDF calculation

Wrong move:

Rounding z-scores to 1 decimal place

Why:

This introduces 10-15% error in final probability values, leading to lost points on AP FRQs

Correct move:

Keep a minimum of 2 decimal places for all z-scores, ideally 3, for intermediate steps

Wrong move:

Using sample standard deviation s for population σ with no justification

Why:

College Board rubrics deduct partial credit for unstated assumptions about population parameters

Correct move:

Explicitly state that you are assuming the given standard deviation is the true population value in your solution

6. Quick Reference Cheatsheet

Concept	Formula / Rule	AP Exam Use Case
Empirical Rule	68% within 1σ, 95% within 2σ, 99.7% within 3σ of μ	Fast MCQ probability estimates
Z-Score	$z = \frac{x - μ}{σ}$	Standardize raw values to the standard normal distribution
Normal CDF	$P (a < X < b) = P (X < b) - P (X < a)$	Find probability a normal variable falls between two values
Inverse Normal	Given left area p, find x where $P (X < x) = p$	Find percentile cutoff values for specified probabilities

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · MCQ
Normal probability calculation
2024 · FRQ 2
Normal model for test scores
2023 · MCQ
Empirical rule application

What's Next

Mastery of the normal distribution is the single most important foundational skill for AP Statistics, as it underpins over 50% of the exam content including sampling distributions, confidence intervals, and hypothesis testing. The z-score logic you learned here will be reused directly to calculate test statistics for z-tests and construct confidence intervals for population means. Before moving forward, ensure you can reliably distinguish between when to use normal CDF vs inverse normal, as this distinction is the most common source of avoidable FRQ point loss. Next, you can extend this knowledge to related core topics that build directly on normal distribution properties.