🎯⭐ INTERACTIVE LESSON

Describing Distributions

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Describing Distributions - Complete Interactive Lesson

Part 1: Shape, Center, Spread

📊 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:

Feature	What It Tells You	Common Measures
Shape	Overall pattern	Symmetric, left-skewed, right-skewed, unimodal, bimodal
Center	Typical value	Mean ( $\bar{x}$ ), Median ( $M$ )
Spread	Variability	Range, IQR, Standard Deviation ( $s$ )

🔑 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

Shape	Description	Visual
Symmetric	Left and right halves are mirror images	Bell-shaped, uniform
Right-skewed	Long tail extends to the RIGHT	Most data clumped left
Left-skewed	Long tail extends to the LEFT	Most data clumped right
Unimodal	One peak	Single hump
Bimodal	Two peaks	Two humps

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

Center

Measure	Formula	Best When
Mean ( $\bar{x}$ )	$\bar{x} = \frac{\sum x_i}{n}$

Key Relationship:

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

Spread

Measure	Formula	Description
Range	$\text{max} - \text{min}$	Simplest; sensitive to outliers
IQR	$Q_3 - Q_1$

Worked Example

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

Feature	Analysis
Shape	Roughly symmetric (approximately bell-shaped)
Center	Median = 4 (5th of 9 values); Mean = $\frac{36}{9} = 4$
Spread	Range = $6 - 2 = 4$ ; IQR =

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 ✅

Part 2: Histograms & Dotplots

📊 Histograms & Dotplots

Part 2 of 7 — Displaying Quantitative Data

Types of Graphs for Quantitative Data

Graph	Description	Best For
Histogram	Bars show frequency of data in intervals (bins)	Large data sets
Dotplot	Dots stacked above a number line	Small data sets; seeing individual values
Stemplot	Stems (leading digits) with leaves (trailing digits)	Moderate data sets; preserving exact values

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

Bin Width	Effect
Too wide (few bins)	Hides patterns in the data

Part 3: Mean vs Median

🔢 Mean vs Median

Part 3 of 7 — Choosing the Right Measure of Center

Mean ( $\bar{x}$ )

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

Part 4: Standard Deviation

📈 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.

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

Part 5: Normal Distribution

🧮 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:

Parameter	Symbol	Role
Mean	$\mu$	Center of the distribution
Standard Deviation	$\sigma$	Controls the width/spread

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

Part 6: Problem-Solving Workshop

🛠️ 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: $\bar{x} = \frac{\sum x_i}{20} = \frac{1458}{20} = 72.9$

Part 7: Review & Applications

🏆 Review & Applications

Part 7 of 7 — Comprehensive Review

Complete Summary

Describing Distributions Checklist

Feature	Measures	Notes
Shape	Symmetric, left/right-skewed, unimodal/bimodal	Direction of TAIL = direction of skew
Center	Mean ( $\bar{x}$ ), Median ( $M$ )	Symmetric → mean; Skewed → median
Spread

Q_3 - Q_1 = 5 - 3 = 2

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

Value	1	2	3	4	5
Dots	•	••	•••	••	•

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

Stemplots (Stem-and-Leaf Plots)

Component	Role
Stem	Leading digit(s)
Leaf	Trailing digit
Key	Tells how to read the values (e.g., 3

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

Stem	Leaves
2	1 3 5
3	1 4 8
4	2 5

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

Reading Graphs — What to Look For

Feature	What to Check
Shape	Symmetric? Skewed? Unimodal? Bimodal?
Center	Where is the "middle" of the data?
Spread	How spread out are the values?
Outliers	Any values far from the rest?
Gaps/Clusters	Are there groups of data with gaps between them?

🔑 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) What is the mode (most frequent value)?

3) What is the range?

Graph Features 🔍

Exit Quiz — Histograms & Dotplots ✅

Property	Detail
Uses every data value	Yes — all values contribute
Sensitive to outliers	Yes — extreme values pull it
Best for	Symmetric distributions
Balance point	The mean is the "balance point" of the distribution

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

Property	Detail
Uses every data value	No — only the position matters
Sensitive to outliers	No — resistant to extreme values
Best for	Skewed distributions or data with outliers

Situation	Use	Why
Symmetric data	Mean or Median	They're approximately equal
Right-skewed data	Median	Mean is inflated by the right tail
Left-skewed data	Median	Mean is deflated by the left tail
Data with outliers	Median	Mean is pulled by outliers
Need to calculate totals	Mean	$\text{Total} = \bar{x} \times n$

Data: 2, 4, 6, 8, 100

Measure	Calculation	Result
Mean	$\frac{2+4+6+8+100}{5} = \frac{120}{5}$	24
Median	Middle (3rd) value of ordered data	6

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 Transformations

Transformation	Effect on Mean	Effect on Median
Add constant $c$ to all values	Add $c$	Add $c$
Multiply all values by $k$	Multiply by $k$	Multiply by $k$
Add an outlier	Pulled toward outlier	Minimal change
Remove an outlier	Moves back toward center	Minimal change

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 ✅

Term	Definition
$x_i - \bar{x}$	Deviation — how far each value is from the mean
$(x_i - \bar{x})^2$	Squared deviation — removes negatives
$\frac{\sum(x_i - \bar{x})^2}{n-1}$
$s = \sqrt{s^2}$	Standard deviation — back in original units

⚠️ 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

Property	Detail
$s \geq 0$ always	Standard deviation can never be negative
$s = 0$	Only when ALL values are identical
Units	Same units as the original data
Sensitive to outliers	Yes — outliers inflate $s$
Affected by transformations	Adding $c$ : $s$ unchanged. Multiplying by $k$ : $s$ multiplied by $

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:

$x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
2	$-4$	16
4	$-2$	4
6	0	0
8	2	4
10	4	16
	Sum	40

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 Transformations on $s$

Transformation	Effect on $s$	Example
Add constant $c$	$s$ unchanged	Data + 10: same spread
Multiply by $k$	$s \times	k

🔑 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 ✅

The Empirical Rule (68-95-99.7 Rule)

For any normal distribution:

Range	Percentage
$\mu \pm 1\sigma$	68% of data
$\mu \pm 2\sigma$	95% of data
$\mu \pm 3\sigma$	99.7% of data

Worked Example 1: Empirical Rule

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

Range	Interval	Percentage
$\mu \pm 1\sigma$	$100 \pm 15 = [85, 115]$	68%
$\mu \pm 2\sigma$	$100 \pm 30 = [70, 130]$	95%
$\mu \pm 3\sigma$	$100 \pm 45 = [55, 145]$	99.7%

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

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

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

z-score	Meaning
$z = 0$	At the mean
$z = 1$	1 SD above the mean
$z = -2$	2 SDs below the mean

🔑 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 $x$ :

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

Calculator Command	Purpose
normalcdf(lower, upper, $\mu$ , $\sigma$ )	Area between two values
invNorm(area, $\mu$ , $\sigma$ )	Value at a given percentile

⚠️ 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 ✅

Median: Average of 10th and 11th values =

\frac{72 + 73}{2} = 72.5

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

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?

(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

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

Term	Formula
Lower fence	$Q_1 - 1.5 \times \text{IQR}$
Upper fence	$Q_3 + 1.5 \times \text{IQR}$
IQR	$Q_3 - Q_1$

🔑 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 ✅

Formula	Expression
Mean	$\bar{x} = \frac{\sum x_i}{n}$
Median position	$(n+1)/2$
Range	max $-$ min
IQR	$Q_3 - Q_1$
Variance	$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$
Standard Deviation	$s = \sqrt{s^2}$
Z-score	$z = \frac{x - \mu}{\sigma}$
Lower fence	$Q_1 - 1.5 \times \text{IQR}$
Upper fence	$Q_3 + 1.5 \times \text{IQR}$

Mean vs Median Decision Guide

If...	Then use...	Because...
Data is symmetric, no outliers	Mean and SD	Mean uses all values; SD pairs with mean
Data is skewed	Median and IQR	Both are resistant to skewness
Data has outliers	Median and IQR	Both are resistant to outliers
You need to find totals	Mean	Total = $\bar{x} \times n$

The Empirical Rule (Normal Distributions Only)

Range	Percentage	Above upper	Below lower
$\mu \pm 1\sigma$	68%	16%	16%
$\mu \pm 2\sigma$	95%	2.5%	2.5%
$\mu \pm 3\sigma$	99.7%	0.15%	0.15%

Key Concepts from Every Part

Part	Topic	Essential Takeaway
1	Shape, Center, Spread	Always describe all three + outliers
2	Histograms & Dotplots	Visual displays for quantitative data
3	Mean vs Median	Skewed → median; Symmetric → mean
4	Standard Deviation	Measures typical distance from mean; $s \geq 0$
5	Normal Distribution	68-95-99.7 rule; z-scores standardize
6	Problem-Solving	Combine concepts; show work; use context

Transformations Summary

Operation	Effect on Center	Effect on Spread
Add $c$	Add $c$	No change
Multiply by $k$	Multiply by $k$	Multiply by $

🔑 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 ✅

Describing Distributions

Dotplots

Stemplots (Stem-and-Leaf Plots)

Reading Graphs — What to Look For

Median ( $M$ )

When to Use Each

Worked Example

Effect of Transformations

Properties of Standard Deviation

Worked Example

Effect of Transformations on $s$

The Empirical Rule (68-95-99.7 Rule)

Worked Example 1: Empirical Rule

Z-Scores

Worked Example 2: Z-Scores

Using the Normal Distribution

Worked Example 2: Normal Distribution Application

Problem

Solution

1.5 × IQR Rule for Outliers

Formula Reference

Mean vs Median Decision Guide

The Empirical Rule (Normal Distributions Only)

Key Concepts from Every Part

Transformations Summary

Describing Distributions

Describing Distributions - Complete Interactive Lesson

Part 1: Shape, Center, Spread

📊 Shape, Center, Spread

The S-C-S Framework

Shape

Center

Spread

Worked Example

Part 2: Histograms & Dotplots

📊 Histograms & Dotplots

Types of Graphs for Quantitative Data

Histograms

Part 3: Mean vs Median

🔢 Mean vs Median

Mean (xˉ\bar{x}xˉ)

Part 4: Standard Deviation

📈 Standard Deviation

Variance and Standard Deviation

Part 5: Normal Distribution

🧮 Normal Distribution & Empirical Rule

The Normal Distribution

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Worked Example 1: Describing a Distribution

Problem

Solution

Part 7: Review & Applications

🏆 Review & Applications

Complete Summary

Describing Distributions Checklist

Dotplots

Stemplots (Stem-and-Leaf Plots)

Reading Graphs — What to Look For

Median (MMM)

When to Use Each

Worked Example

Effect of Transformations

Properties of Standard Deviation

Worked Example

Effect of Transformations on sss

The Empirical Rule (68-95-99.7 Rule)

Worked Example 1: Empirical Rule

Z-Scores

Worked Example 2: Z-Scores

Using the Normal Distribution

Worked Example 2: Normal Distribution Application

Problem

Solution

1.5 × IQR Rule for Outliers

Formula Reference

Mean vs Median Decision Guide

The Empirical Rule (Normal Distributions Only)

Key Concepts from Every Part

Transformations Summary

Mean ( $\bar{x}$ )

Median ( $M$ )

Effect of Transformations on $s$