Describing Distributions

📊 Shape, Center, Spread

Part 1 of 7 — The Three Key Features of Any Distribution

The S-C-S Framework

Every distribution of quantitative data should be described using three features:

🔑 AP Tip: On free-response questions, you MUST address all three features (shape, center, spread) AND mention any outliers. Missing any one costs points.

Shape

⚠️ Common Mistake: The direction of skewness is the direction of the TAIL, not where most data is concentrated.

Center

Key Relationship:

• Right-skewed → mean > median (mean pulled toward tail)
• Left-skewed → mean < median (mean pulled toward tail)
• Symmetric → mean $\approx$ median

Spread

Worked Example

Data: 2, 3, 3, 4, 4, 4, 5, 5, 6

AP-Style Description: "The distribution of values is roughly symmetric and unimodal with a center (median) of 4. The spread is moderate with an IQR of 2 and a range of 4. There are no apparent outliers."

Shape, Center, Spread Concepts 🎯

Calculations 🧮

Data: 1, 3, 5, 7, 9, 11, 13

1) What is the median?

2) What is the range?

3) What is the mean?

Choosing the Right Measure 🔍

Exit Quiz — Shape, Center, Spread ✅

📊 Histograms & Dotplots

Part 2 of 7 — Displaying Quantitative Data

Types of Graphs for Quantitative Data

Histograms

A histogram divides values into equal-width bins and shows the frequency (count) or relative frequency (proportion) of each bin.

Key Features:

• Bars touch (no gaps between adjacent bins)
• x-axis: quantitative variable (bins)
• y-axis: frequency or relative frequency
• Bin width affects appearance — too few bins hide detail, too many create noise

Dotplots

Each data value is represented by a dot above a number line. Dots stack when values repeat.

Example: Data: 1, 2, 2, 3, 3, 3, 4, 4, 5

• Shape: symmetric
• Center: 3
• Spread: range = 4
• No obvious outliers

Stemplots (Stem-and-Leaf Plots)

Example: Ages: 21, 23, 25, 31, 34, 38, 42, 45

Key: 2|1 = 21 years

Back-to-back stemplots compare two groups using the same stems.

Reading Graphs — What to Look For

🔑 AP Tip: When comparing two distributions, use comparative language: "Distribution A is more spread out than Distribution B" rather than describing each separately.

Graph Interpretation 🎯

Dotplot Reading 🧮

Dotplot frequencies: Value 1→1 dot, 2→3 dots, 3→4 dots, 4→2 dots, 5→1 dot

1) Total number of data points?

2) Which value occurs most often?

3) What is the range?

Graph Features 🔍

Exit Quiz — Histograms & Dotplots ✅

🔢 Mean vs Median

Part 3 of 7 — Choosing the Right Measure of Center

Mean ($\bar{x}$)

$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

Median ($M$)

Procedure:

1. Order all data from smallest to largest
2. If $n$ is odd: median = middle value (position $\frac{n+1}{2}$)
3. If $n$ is even: median = average of the two middle values

When to Use Each

Worked Example

Data: 2, 4, 6, 8, 100

The outlier 100 pulls the mean up to 24, but the median (6) better represents the typical value.

🔑 AP Tip: When asked "which measure of center is more appropriate," always explain WHY. Connect your answer to the shape of the distribution or the presence of outliers.

Effect of Adding or Multiplying by a Constant

Mean vs Median Concepts 🎯

Calculations 🧮

1) Mean of: 8, 12, 16, 20, 24

2) Median of: 3, 7, 9, 15, 21, 25

3) Mean of: 1, 2, 3, 4, 100

Choosing the Right Measure 🔍

Exit Quiz — Mean vs Median ✅

📈 Standard Deviation

Part 4 of 7 — Measuring Spread from the Mean

Variance and Standard Deviation

The standard deviation ($s$) measures the typical distance of data values from the mean.

$s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}$

⚠️ Why $n - 1$? We divide by $n - 1$ (not $n$) because the sample mean $\bar{x}$ is estimated from the data. This gives a better estimate of the population SD. The value $n - 1$ is called the degrees of freedom.

Properties of Standard Deviation

Worked Example

Data: 2, 4, 6, 8, 10. Find $s$.

Step 1: Mean $\bar{x} = \frac{2+4+6+8+10}{5} = \frac{30}{5} = 6$

Step 2: Deviations and squared deviations:

Step 3: Variance: $s^2 = \frac{40}{5-1} = \frac{40}{4} = 10$

Step 4: Standard deviation: $s = \sqrt{10} \approx 3.16$

Interpretation: The values are typically about 3.16 units from the mean of 6.

Effect of Adding or Multiplying by a Constant on $s$

🔑 AP Tip: Adding a constant shifts all values equally, so the spread doesn't change. Multiplying stretches or compresses the data, changing the spread.

Standard Deviation Concepts 🎯

SD Calculations 🧮

1) Data: 3, 5, 7. Mean = 5. What is the variance $s^2$? (Hint: sum of squared deviations ÷ ($n - 1$))

2) What is the standard deviation $s$ of the data above?

3) Data: 10, 10, 10. Standard deviation $s$ = ?

SD Properties 🔍

Exit Quiz — Standard Deviation ✅

🧮 Normal Distribution & Empirical Rule

Part 5 of 7 — The Bell Curve

The Normal Distribution

A normal distribution is a symmetric, bell-shaped curve completely described by two parameters:

Notation: $X \sim N(\mu, \sigma)$

The Empirical Rule (68-95-99.7 Rule)

For any normal distribution:

Worked Example 1: Empirical Rule

IQ scores: $\mu = 100$, $\sigma = 15$

So about 68% of IQ scores fall between 85 and 115.

Z-Scores

A z-score standardizes any value by telling how many SDs it is from the mean:

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

🔑 Key Insight: Z-scores let you compare values from DIFFERENT distributions. A student with $z = 1.5$ on a math test did relatively better than a student with $z = 0.8$ on an English test.

Worked Example 2: Z-Scores

SAT Math: $\mu = 500$, $\sigma = 100$. A student scores 680.

$z = \frac{680 - 500}{100} = \frac{180}{100} = 1.8$

This score is 1.8 standard deviations above the mean.

Using the Normal Distribution

To find the percentage of values below a value $x$:

1. Calculate $z = \frac{x - \mu}{\sigma}$
2. Use the z-table (or calculator: normalcdf) to find the area

⚠️ AP Tip: Always show the z-score calculation AND draw a sketch of the normal curve with the area shaded.

Normal Distribution Concepts 🎯

Z-Score Calculations 🧮

Distribution: $\mu = 200$, $\sigma = 25$

1) What is the z-score for $x = 250$?

2) What value has $z = -1$?

3) The 68% interval is $[\text{lower}, \text{upper}]$. What is the lower bound?

Normal Curve Properties 🔍

Exit Quiz — Normal Distribution ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — Putting It All Together

Worked Example 1: Describing a Distribution

Problem

A teacher records quiz scores for 20 students: 45, 52, 55, 60, 62, 65, 67, 68, 70, 72, 73, 75, 76, 78, 80, 82, 85, 88, 90, 95

Describe this distribution completely.

Solution

Shape: The distribution is roughly symmetric with a slight left skew (the lower scores stretch a bit further from center).

Center:

• Mean: $\bar{x} = \frac{\sum x_i}{20} = \frac{1438}{20} = 71.9$
• Median: Average of 10th and 11th values = $\frac{72 + 73}{2} = 72.5$

Spread:

• Range: $95 - 45 = 50$
• IQR: $Q_1 = 63.5$, $Q_3 = 81$, IQR $= 81 - 63.5 = 17.5$

Outliers: Using the 1.5 × IQR rule:

• Lower fence: $63.5 - 1.5(17.5) = 37.25$ → no values below this
• Upper fence: $81 + 1.5(17.5) = 107.25$ → no values above this
• No outliers

AP-Style Response: "The distribution of quiz scores is roughly symmetric and unimodal with a center (median) of 72.5 points. The scores have moderate spread with an IQR of 17.5 points and a range of 50 points. There are no apparent outliers."

Worked Example 2: Normal Distribution Application

Problem

Heights of adult women are normally distributed with $\mu = 64.5$ inches and $\sigma = 2.5$ inches.

(a) What percentage of women are taller than 69.5 inches?

(b) What height corresponds to the 16th percentile?

Solution

(a) $z = \frac{69.5 - 64.5}{2.5} = \frac{5}{2.5} = 2.0$

69.5 inches is exactly $\mu + 2\sigma$. By the Empirical Rule, 95% of data falls within $\mu \pm 2\sigma$, so 5% is outside this range, and 2.5% is above $\mu + 2\sigma$.

Answer: About 2.5% of women are taller than 69.5 inches.

(b) The 16th percentile means 16% of data is below this value. By the Empirical Rule:

• 50% is below the mean
• 34% is between $\mu - \sigma$ and $\mu$ (half of 68%)
• So 16% is below $\mu - \sigma$

$\mu - \sigma = 64.5 - 2.5 = 62.0$ inches

Answer: The 16th percentile is approximately 62.0 inches.

1.5 × IQR Rule for Outliers

$\text{Outlier if } x < Q_1 - 1.5 \times \text{IQR} \text{ or } x > Q_3 + 1.5 \times \text{IQR}$

🔑 AP Tip: On free-response questions, SHOW the fence calculations when identifying outliers. Don't just say "there are outliers" — prove it mathematically.

Workshop Practice 🎯

Calculations 🧮

1) $\mu = 100$, $\sigma = 15$. What percentage of data is between 85 and 115? (answer as whole number)

2) $Q_1 = 30$, $Q_3 = 50$. What is the IQR?

3) Using the values above, what is the lower fence for outliers?

Decision Making 🔍

Exit Quiz — Problem-Solving Workshop ✅

🏆 Review & Applications

Part 7 of 7 — Comprehensive Review

Complete Summary

Describing Distributions Checklist

Formula Reference

Mean vs Median Decision Guide

The Empirical Rule (Normal Distributions Only)

Key Concepts from Every Part

Changing Units: Adding or Multiplying by a Constant

🔑 Final AP Tip: On every free-response question, use CONTEXT. Don't just say "the distribution is right-skewed with a center of 72." Say "the distribution of quiz scores is right-skewed with a median of 72 points."

Comprehensive Review 🎯

Formula Application 🧮

1) Data: 10, 15, 20, 25, 30. What is the mean?

2) $\mu = 80$, $\sigma = 6$. What z-score corresponds to $x = 92$?

3) $Q_1 = 25$, $Q_3 = 45$. What is the upper fence for outliers?

Concept Connections 🔍

Final Exam — Describing Distributions ✅