AP Statistics Topic 2.11: The Normal Distribution

01 · Start here

The concept in 60 seconds

A normal distribution is a continuous, symmetric, single-peaked bell curve. Its mean tells you where the center is; its standard deviation tells you how spread out the values are.

For a continuous model, probability is an area under the curve over a region. A taller point on the curve does not itself give the probability of that exact value.

Two directions, two questions:
“What proportion finishes before $44$ minutes?” gives a value and asks for an area.
“How quickly do the fastest $10\%$ finish?” gives an area and asks for a value.

We will use an original fictional teaching model: the time $X$ , in minutes, for $1$ student to finish a practice activity is modeled as normal with mean $50$ minutes and SD $8$ minutes. The normal assumption is supplied for this lesson; it is not a claim about real student records.

Quick check: is every symmetric distribution normal?

No. A symmetric uniform distribution is flat, and some symmetric distributions have multiple peaks. Symmetry alone does not justify a normal model. Check the shape and the information supplied by the question.

02 · Describe the model before using it

Shape, mean and standard deviation

Continuous random variable

A measurement can take any value within its modeled range, including decimals. Completion times such as $47.2$ and $47.25$ minutes are possible.

Contrast this with a binomial count, which takes integer values.

Mean $\mu$ : the center

$\mu$ locates the center of the normal curve. Mean, median and mode all coincide there.

Here $\mu = 50$ minutes; $\frac12\text{ of the model area}$ is on each side.

SD $\sigma$ : the spread

$\sigma$ is positive and uses the same units as $X$ . A larger SD gives a wider, lower curve; a smaller SD gives a narrower, taller curve.

Here $\sigma = 8$ minutes. Variance $\sigma^2 = 64 \mathrm{min}^{2}$ .

$\text{Total area: }1$

All possible outcomes together have probability $1$ . Increasing the curve’s width does not increase its total probability.

Curve height measures density; region area measures probability.

Write the model clearly as $X \sim \operatorname{Normal}(\mu = 50,\sigma = 8)$ . If your course uses $\mathcal{N}(\mu , \sigma )$ , this is $\mathcal{N}(50, 8)$ . Some texts use $\mathcal{N}(\mu , \sigma^2)$ , so always label whether the second number is the SD or the variance.

Visual guide 1: what the two parameters change

Solid and dashed curves distinguish the models. Both panels use the same numerical scales. Every complete normal curve has area $1$ ; the display shows a finite portion of its tails.

What can justify a normal model?

Use the normal assumption when a problem gives it. When you are assessing data, a roughly symmetric, unimodal bell-shaped distribution without strong skew or extreme outliers can support an approximately normal model. A small data set can look irregular even when a normal model is reasonable; shape is evidence to assess, not automatic proof.

A normal curve has tails that extend indefinitely. A real measurement may have physical limits, so a normal model is an approximation to reality and should be suitable for the range being studied.

Keep the variable straight: This lesson models $1$ student’s completion time. Its SD is $8$ minutes. Do not replace it with $\frac{8}{\sqrt{n}}$ (where $n$ is the sample size); the distribution of a sample mean is a different topic.

Quick check: what are the mean and SD of the standard normal distribution?

The standard normal distribution has mean $0$ and SD $1$ . It describes standardized values $Z$ rather than completion times in minutes.

03 · Estimate from whole-SD landmarks

The $68\%\text{–}95\%\text{–}99.7\%$ rule

For a normal distribution, about $68\%$ of observations lie within $1$ SD of the mean, about $95\%$ within $2$ SDs, and about $99.7\%$ within $3$ SDs. These are useful approximations.

Visual guide 2: the three central empirical-rule bands

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The Normal Distribution

By the end of this lesson, you should be able to:

The concept in 60 seconds