The Normal distribution is one of the most important probability models in statistics. It describes many real-world phenomena: heights, test scores, measurement errors, and more.
Interactive: Adjust $\mu$ and $\sigma$ to see how the Normal curve changes shape and location
Figure 3.1 — Normal Distribution $N(\mu, \sigma)$ — Shape and Location
For any Normal distribution $N(\mu, \sigma)$, the Empirical Rule gives the percentage of data within 1, 2, and 3 standard deviations of the mean. (Reviewed from Chapter 2, now applied to Normal models.)
68% of values lie within $\mu \pm \sigma$
95% of values lie within $\mu \pm 2\sigma$
99.7% of values lie within $\mu \pm 3\sigma$
Equivalently: 16% are below $\mu - \sigma$ (and 16% above $\mu + \sigma$); 2.5% are below $\mu - 2\sigma$; 0.15% are below $\mu - 3\sigma$.
Women's heights are approximately $N(64.5, 2.5)$ inches (mean 64.5 in, SD 2.5 in).
(a) What percent of women are between 59.5 and 69.5 inches?
59.5 = 64.5 − 2(2.5) = $\mu - 2\sigma$ and 69.5 = $\mu + 2\sigma$ → 95%
(b) What percent are shorter than 57 inches?
57 = 64.5 − 3(2.5) = $\mu - 3\sigma$. Below $\mu - 3\sigma$: 0.3%/2 = 0.15%
(c) Between what heights are the middle 68% of women?
$\mu \pm \sigma = 64.5 \pm 2.5$ → between 62 and 67 inches
SAT Math scores are $N(530, 115)$. (a) What percent score between 185 and 875? (b) What percent score above 760? (c) Between what scores are the middle 95%?
The standard Normal distribution is the special case $N(0,1)$ — mean 0, standard deviation 1. We convert any Normal value to a z-score to use the standard Normal table.
$$z = \frac{x - \mu}{\sigma}$$
The z-score tells us how many standard deviations $x$ is from $\mu$. After standardizing, all Normal distributions become $N(0,1)$, so we can use one table.
The standard Normal table gives the proportion of values at or below a given z-score (the area to the left of $z$ under the $N(0,1)$ curve). This proportion equals the percentile of the value.
A partial z-table (positive z-scores):
| z | .00 | .01 | .02 | .03 | .04 | .05 | .06 | .07 | .08 | .09 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0.0 | .5000 | .5040 | .5080 | .5120 | .5160 | .5199 | .5239 | .5279 | .5319 | .5359 |
| 0.5 | .6915 | .6950 | .6985 | .7019 | .7054 | .7088 | .7123 | .7157 | .7190 | .7224 |
| 1.0 | .8413 | .8438 | .8461 | .8485 | .8508 | .8531 | .8554 | .8577 | .8599 | .8621 |
| 1.5 | .9332 | .9345 | .9357 | .9370 | .9382 | .9394 | .9406 | .9418 | .9429 | .9441 |
| 2.0 | .9772 | .9778 | .9783 | .9788 | .9793 | .9798 | .9803 | .9808 | .9812 | .9817 |
| 2.5 | .9938 | .9940 | .9941 | .9943 | .9945 | .9946 | .9948 | .9949 | .9951 | .9952 |
| 3.0 | .9987 | .9987 | .9987 | .9988 | .9988 | .9989 | .9989 | .9989 | .9990 | .9990 |
AP Exam Tip: The table gives area to the left. To find area to the right, compute $1 - \text{table value}$. To find area between two z-scores, compute (larger area) − (smaller area). Always draw a sketch first.
IQ scores are $N(100, 15)$. What proportion of people have IQ above 118?
Step 1: $\mu = 100$, $\sigma = 15$, want $P(X > 118)$.
Step 2: (Sketch — shade right tail above 118.)
Step 3 — Standardize: $z = (118 - 100)/15 = 18/15 = 1.20$
Step 4 — Table: $P(Z \le 1.20) = 0.8849$
Step 5: $P(X > 118) = 1 - 0.8849 = \mathbf{0.1151}$
About 11.5% of people have IQ above 118.
Adult male blood pressure is $N(125, 14)$ mmHg. What percent have pressure between 111 and 139?
Standardize both:
$z_1 = (111 - 125)/14 = -14/14 = -1.00$
$z_2 = (139 - 125)/14 = 14/14 = 1.00$
Area between: $P(-1 \le Z \le 1) = P(Z \le 1) - P(Z \le -1)$
$= 0.8413 - 0.1587 = \mathbf{0.6826}$
About 68.3% — confirming the Empirical Rule ($\mu \pm \sigma$).
Interactive: Find the shaded area — drag the left and right z-score bounds
Figure 3.2 — Normal Probability: Shaded Area Between Two Z-Scores
AP Calculus scores are approximately $N(2.9, 1.1)$ (on a 1–5 scale). What score is at the 90th percentile?
Step 1: Need $x$ such that $P(X \le x) = 0.90$.
Step 2: Look up 0.90 in z-table → $z \approx 1.28$.
Step 3 — Unstandardize: $x = 2.9 + (1.28)(1.1) = 2.9 + 1.408 = \mathbf{4.31}$
A score of about 4.3 is at the 90th percentile for AP Calculus.
Heights of adult males are $N(70, 3)$ inches. (a) What percent are between 64 and 73 inches? (b) A man is at the 25th percentile — what is his height?
Interactive: Given a percentile (shaded area), find the corresponding z-score — adjust the area slider
Figure 3.3 — Inverse Normal: Finding a Value from a Percentile
We can assess whether a dataset is approximately Normal using:
AP Exam Tip: On the AP exam, you often need to check the Large Counts or Normality condition before performing inference. For sample means, the Central Limit Theorem (Ch 9) handles this. For now, state "approximately Normal" only when the data graph looks symmetric and bell-shaped, or when the problem states Normality.
Scores on a standardized test are $N(500, 100)$. Using the Empirical Rule, find the percent of scores (a) between 300 and 700, (b) above 600, (c) below 400.
Find the z-score for each value, given $N(85, 12)$: (a) $x = 97$ (b) $x = 73$ (c) $x = 85$
Newborn weights are $N(7.5, 1.1)$ lb. What percent of newborns weigh less than 6 pounds?
Lifetime of a light bulb is $N(1200, 80)$ hours. Find the probability a bulb lasts between 1100 and 1350 hours.
AP Biology scores are $N(2.8, 1.0)$. What score corresponds to the (a) 75th percentile? (b) 10th percentile?
Resting heart rate is $N(72, 10)$ bpm. (a) What percent have rate above 90? (b) Between what rates are the middle 90% of people?
Two students compare scores. Student A got 82 on a test with $N(75, 8)$. Student B got 72 on a test with $N(65, 5)$. Who performed better relative to their class?
(AP Free Response Style) Annual rainfall in a city is approximately $N(32, 5)$ inches. (a) Find $P(25 \le X \le 40)$. (b) The city issues a drought warning when annual rainfall is below the 15th percentile. What is the drought warning threshold?
Bell-shaped, symmetric, described by mean $\mu$ and standard deviation $\sigma$. Notation: $N(\mu, \sigma)$. The curve extends infinitely in both directions.
In any Normal distribution: ~68% within $\pm 1\sigma$, ~95% within $\pm 2\sigma$, ~99.7% within $\pm 3\sigma$ of the mean.
$z = \dfrac{x - \mu}{\sigma}$ — number of standard deviations from the mean. Standardizes values to $N(0,1)$.
Table of $P(Z \leq z)$ for $Z \sim N(0,1)$. Use to find areas (probabilities) by looking up the z-score.