Normal Distributions

📊 The Normal Distribution

Part 1 of 7 — Bell Curves and the Empirical Rule

The Normal Distribution

The normal distribution is the most important distribution in statistics. It is:

• Symmetric and bell-shaped
• Described by two parameters: mean $\mu$ and standard deviation $\sigma$
• Notation: $X \sim N(\mu, \sigma)$

The Empirical Rule (68-95-99.7)

For any normal distribution:

Example: IQ scores follow $N(100, 15)$

• 68% of scores between $100 \pm 15 = [85, 115]$
• 95% between $100 \pm 30 = [70, 130]$
• 99.7% between $100 \pm 45 = [55, 145]$

🔑 The Empirical Rule gives a quick estimate for normal data. For exact probabilities, use z-scores.

Normal Distribution Check 🎯

Empirical Rule Calculations 🧮

Adult male heights follow $N(70, 3)$ inches.

1) What percentage of men are between 67 and 73 inches tall?

2) What percentage are shorter than 64 inches? (Hint: 64 = 70 − 2(3))

3) Between what two heights do the middle 99.7% of men fall? Give the upper bound.

📏 Z-Scores and the Standard Normal

Part 2 of 7 — Standardizing Values

The Z-Score Formula

A z-score tells you how many standard deviations a value is from the mean:

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

The Standard Normal Distribution

When we standardize: $Z \sim N(0, 1)$

This allows us to use one table (or calculator) for all normal distributions.

Example: Heights $\sim N(70, 3)$. A person is 76 inches tall. $z = \frac{76 - 70}{3} = 2$ They are 2 standard deviations above the mean.

Using the Z-Table

The z-table gives $P(Z \leq z)$ — the area to the left of $z$.

🔑 Always sketch the normal curve, shade the region, then calculate.

Z-Score Practice 🎯

Z-Score Calculations 🧮

ACT scores follow $N(21, 5)$.

1) Find the z-score for a student who scored 31. (Give as a whole number)

2) Find the z-score for a student who scored 16.

3) A student has $z = -0.4$. What was their ACT score?

🔢 Normal Probability Calculations

Part 3 of 7 — Finding Areas and Percentiles

Forward Problems: X → Z → Probability

Given a value $x$, find the probability:

1. Compute $z = \frac{x - \mu}{\sigma}$
2. Look up $P(Z \leq z)$ in the z-table
3. Adjust for the direction (left tail, right tail, between)

Example: Scores $\sim N(500, 100)$. Find $P(X > 650)$.

• $z = \frac{650 - 500}{100} = 1.5$
• $P(Z \leq 1.5) = 0.9332$
• $P(X > 650) = 1 - 0.9332 = 0.0668 = 6.68\%$

Backward Problems: Probability → Z → X

Given a percentile, find the value:

1. Find the z-score from the table that matches the given probability
2. Solve for $x = \mu + z\sigma$

Example: What score is at the 90th percentile if $\mu = 500, \sigma = 100$?

• 90th percentile → $z = 1.28$ (from table: $P(Z \leq 1.28) = 0.8997 \approx 0.90$)
• $x = 500 + 1.28(100) = 628$

🔑 "Top 10%" = 90th percentile. "Bottom 25%" = 25th percentile.

Normal Calculations 🎯

Finding Percentiles 🧮

Baby weights at birth follow $N(7.5, 1.2)$ lbs.

1) What z-score corresponds to a baby weighing 9.9 lbs?

2) Using $P(Z \leq 2) = 0.9772$, what percent of babies weigh less than 9.9 lbs? (Express as a number, e.g., 97.72)

3) The 84th percentile has $z \approx 1$. What is the 84th percentile weight? (in lbs, one decimal)

📈 Assessing Normality

Part 4 of 7 — Is a Normal Model Reasonable?

Why Check?

A normal curve can model a distribution of data only if the data have a roughly normal shape. Later, inference procedures for means also ask whether the sample data could reasonably come from a normal population.

Check the Shape with a Graph

Make a histogram, dotplot, stemplot or boxplot of the data and look for:

🔑 No real data set is perfectly normal. We look for "close enough": roughly symmetric and unimodal with no outliers.

The AP exam checks normality with graphs and the empirical rule; normal probability plots are not part of the course.

Normality Assessment 🎯

Check with the Empirical Rule

Compare the actual percent of data within 1, 2 and 3 standard deviations of the mean with 68%, 95% and 99.7%.

Example: 40 test scores have $\bar{x} = 50$ and $s = 8$.

The percents are close to 68–95–99.7, so a normal model is reasonable.

The Sample Data Condition (Preview of Unit 4)

For t-procedures, the population distribution should be approximately normal or $n \geq 30$. With a small sample, check a graph of the sample data: it should be free from strong skewness and outliers.

Empirical Rule Check 🧮

A sample of 50 values has $\bar{x} = 20$ and $s = 4$. There are 33 values between 16 and 24, 48 values between 12 and 28, and all 50 between 8 and 32.

1) Percent within 1 SD of the mean?

2) Percent within 2 SDs?

3) Percent within 3 SDs?

Sample Data Condition 🔍

➕ Intervals, Percentiles and Comparing Relative Position

Part 5 of 7 — Working with Areas in Both Directions

Setting Up the Inequality

Middle Intervals

Example: Battery life is $N(500, 40)$ hours. Find the middle 90%.

• Tails: 10% outside, so 5% in each tail.
• $z = \pm 1.645$ (the 5th and 95th percentiles of $Z$).
• Interval: $500 \pm 1.645(40) = 434.2$ to $565.8$ hours.

The Highest $p$%

Example: Which battery life marks the top 15%?

• 85% of the area is to the left: $z \approx 1.04$.
• $x = 500 + 1.04(40) = 541.6$ hours.

🔑 Always translate "top", "bottom", "middle" into the area to the LEFT before using a table or invNorm.

Intervals and Cutoffs 🎯

Comparing Relative Position

Percentiles (or z-scores) let you compare values from different normal distributions.

Example: Ana scores 82 on a test that is $N(74, 6)$. Ben scores 88 on a test that is $N(80, 10)$.

Ben's raw score is higher, but Ana did better relative to her group.

⚠️ When smaller is better (race times, golf scores), a more negative z-score is the better performance.

Relative Position 📊

Intervals and Percentiles 🧮

Exam scores are $N(500, 100)$.

1) Lower cutoff of the middle 95% of scores? (use $z = 1.96$)

2) Score at the 30th percentile? (use $z = -0.52$, nearest whole number)

3) A student scores 650. At what percentile is that score? (nearest whole number)

🛠️ Normal Distribution Workshop

Part 6 of 7 — Comprehensive Practice

Strategy for Normal Distribution Problems

1. Identify $\mu$ and $\sigma$ from the problem
2. Sketch the curve and shade the desired region
3. Standardize using $z = (x - \mu)/\sigma$
4. Use the table or calculator to find probabilities
5. For percentiles: work backward from probability to z to x

Worked Example

Problem: A machine fills cereal boxes with $\mu = 368$ g and $\sigma = 4$ g. What proportion of boxes have less than 360 g?

Solution:

1. $z = (360 - 368)/4 = -8/4 = -2$
2. $P(Z \leq -2) = 0.0228$
3. About $2.28\%$ of boxes are underfilled.

Follow-up: What weight is exceeded by 90% of boxes?

1. "Exceeded by 90%" means 10th percentile (10% are below)
2. $z = -1.28$ (from table)
3. $x = 368 + (-1.28)(4) = 368 - 5.12 = 362.88$ g

Workshop Problems 🎯

📋 Normal Distribution Review

Part 7 of 7 — Summary and Applications

Key Formulas

Common z-Values to Know

Final Review 🎯

Mastery Review 🔽