Lesson 2: Sampling & Study Design

Cal

Reese

DMath Basketball!!

Math vs TBD

6-2

Admin

Don’t Forget

Mandatory and Important Briefings

A Note on AI

  • AI is welcome and encouraged on WebAssign. Screenshot the question… maybe.
  • But when WPR time comes - no AI. So use it as a tool for learning, not a crutch.
  • WebAssign is 15% of the course. You should get most/all those points.
  • Don’t get those points at the peril of your WPR grades - that’s 60% of the course.
  • WebAssign is forgiving: 5 tries per sub-question.
  • Don’t forget documentation. Just tell me what you did.

Graded Assignments

Assignment Points
WebAssign Homework 150
WPR I 175
WPR II 175
Exploratory Data Analysis 25
Tech Report 125
Project Presentation 50
TEE 300
Total 1000

Request a Vantage Account

ImportantAction Required

Before we continue, everyone needs to request an account on Vantage - the Army’s data science platform where we’ll be using R this semester.

  1. Go to https://vantage.army.mil/
  2. Click Request Account
  3. Fill out the required information
  4. For Commander email, use: jonathan.l.day3.mil@army.mil
  5. Wait for approval (may take 1-2 days)

We’ll be using Vantage for the project throughout the course.

Lesson 1 Review

The Big Picture

Why do we collect data? Because we want to learn about something bigger than what we can directly observe.

  • Population: The entire group we want to learn about
  • Sample: The subset we actually observe
  • Process: An ongoing mechanism that generates data over time

The whole course builds on this: we use samples to make claims about populations. That’s inference.

Parameters vs Statistics

Population Sample
What we have Usually unknown Observable data
Notation Greek letters (\(\mu\), \(\sigma\), \(\pi\)) Latin letters (\(\bar{x}\), \(s\), \(\hat{p}\))

A statistic estimates a parameter. This is the foundation of Blocks 2 and 3.

Types of Variables

  • Categorical: Labels or categories (e.g., branch, major, yes/no)
  • Quantitative: Numbers with meaningful arithmetic (e.g., height, GPA, time)

Why does this matter? The type of variable determines:

  • How you summarize it (proportions vs means)
  • How you visualize it (bar charts vs histograms)
  • Which inference method you use (z-test for proportions vs t-test for means)

Lesson 2 Content

Objectives

  • Construct and interpret stem-and-leaf displays
  • Create and interpret dotplots
  • Build and analyze histograms and frequency distributions
  • Describe distributions in terms of shape, center, spread, and outliers

Required Reading

Devore, Section 1.2: Pictorial and Tabular Methods in Descriptive Statistics

Why Visualize Data?

Before calculating any numbers, we should look at the data. Visual displays help us:

  • Identify the shape of the distribution
  • Find a typical value (center)
  • See how much variability exists (spread)
  • Spot outliers and gaps
  • Check for symmetry or skewness

Four Types of Displays

Comparing the Four Display Types

Display Description Best Sample Size Preserves Exact Values?
Stem-and-leaf Split each number into stem (leading digits) and leaf (trailing digit) Small (n < 50) Yes
Dotplot Place a dot for each observation on a number line; stack repeated values Small (n < 30) Yes
Histogram (Frequency) Divide range into bins; bar height = count in each bin Any size No
Histogram (Relative Freq) Divide range into bins; bar height = proportion in each bin Any size No

What Can You Learn From Each?

Display Shape Center Spread Outliers Exact Values Compare Groups
Stem-and-leaf Yes Yes Yes Yes Yes Limited
Dotplot Yes Yes Yes Yes Yes Good
Histogram (Frequency) Yes Approximate Yes Yes No Difficult
Histogram (Relative Freq) Yes Approximate Yes Yes No Best

When to Use Each Display

Display Use When… Example
Stem-and-leaf Small dataset, want to preserve exact values Quiz scores for your 18-person section
Dotplot Small dataset with repeated values Number of absences per cadet
Histogram (Frequency) Larger dataset, want raw counts per bin APFT scores for an entire battalion
Histogram (Relative Freq) Comparing groups of different sizes Run times: Company A (120 soldiers) vs Company B (85 soldiers)

Describing Distributions

The Four Key Features

When describing any distribution, always address:

  1. Shape: Symmetric, skewed left, skewed right, unimodal, bimodal?
  2. Center: Where is the “typical” value?
  3. Spread: How much variability is there?
  4. Outliers: Any unusual observations?
TipMemory Aid

S-C-S-O: Shape, Center, Spread, Outliers

Shape: Symmetry and Skewness

Shape Description Relationship
Symmetric Left and right sides are mirror images Mean ≈ Median
Skewed Right Long tail extends to the right Mean > Median
Skewed Left Long tail extends to the left Mean < Median

Examples:

  • Skewed right: Income, home prices
  • Skewed left: Age at retirement, exam scores with a ceiling

Shape: Modality

  • Unimodal: One peak (most common)
  • Bimodal: Two distinct peaks (may indicate two subgroups)
  • Multimodal: Multiple peaks
  • Uniform: Roughly flat, no clear peak

Identifying Outliers

WarningOutlier

An observation that falls far from the rest of the data. Could indicate:

  • Data entry error
  • Measurement error
  • A genuinely unusual observation
  • A different population

Always investigate outliers - don’t automatically remove them!

Example: Cadet Heights

Let’s use simulated height data for 30 cadets to demonstrate all four display types.

Heights (inches): 65, 66, 67, 67, 68, 68, 69, 69, 70, 70, 70, 70, 70, 70, 70, 70, 70, 71, 71, 71, 71, 71, 71, 73, 73, 73, 73, 74, 74, 75

Stem-and-Leaf Display

Stem | Leaves
-----+--------
  6 | 5 6 7 7 8 8 9 9
  7 | 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 3 3 3 3 4 4 5

Reading it: The row “6 | 5 6 7 7 8 8 9 9” represents heights of 65, 66, 67, 67, 68, 68, 69, 69 inches.

Interpreting with S-C-S-O:

  • Shape: Roughly symmetric (similar number of leaves on each stem), unimodal
  • Center: The 7 stem has many more leaves, so center is around 70 inches
  • Spread: Values range from 65 to 75 inches (range = 10 inches)
  • Outliers: No values stand apart from the rest

Unique advantage: We can recover every exact data value! We know there are exactly two cadets who are 67 inches tall.

Dotplot

Interpreting with S-C-S-O:

  • Shape: Roughly symmetric with one clear peak (unimodal); dots cluster in the middle
  • Center: The tallest stack of dots is at 70 inches - this is our center
  • Spread: Dots extend from about 65 to 75 inches
  • Outliers: No dots are isolated far from the others

Unique advantage: Easy to see exact values AND repeated values (stacked dots). The height of 70 inches appears 9 times - we can count the dots!

Histogram (Frequency)

Interpreting with S-C-S-O:

  • Shape: Roughly symmetric (bars rise then fall), unimodal (one peak)
  • Center: Tallest bar is at 70-72 inches, so center is around 70-71
  • Spread: Data spans from about 64 to 76 inches
  • Outliers: No bars are isolated from the main distribution

Unique advantage: Shows counts - we can say “15 cadets are between 70-72 inches tall.” Good for understanding raw numbers.

Histogram (Relative Frequency)

Interpreting with S-C-S-O:

  • Shape: Same as frequency histogram - symmetric, unimodal
  • Center: Peak is at 70-72 inches
  • Spread: Same range, 64 to 76 inches
  • Outliers: None visible

Unique advantage: Shows proportions - we can say “about 50% of cadets are between 70-72 inches tall.” Essential for comparing groups of different sizes (e.g., comparing this section of 30 to another section of 18).

Side-by-Side Comparison

Describing the Cadet Heights Distribution

Using S-C-S-O:

  • Shape: Roughly symmetric, unimodal
  • Center: Around 70 inches
  • Spread: Ranges from about 64 to 76 inches
  • Outliers: None obvious

Practice Problem

The following data represents the number of hours cadets studied for a WPR:

3, 5, 4, 8, 6, 5, 12, 4, 5, 7, 6, 5, 4, 6, 5, 3, 5, 6, 4, 5

  1. Create a stem-and-leaf display for this data
  2. Describe the distribution (shape, center, spread, outliers)
  3. What is the relative frequency of cadets who studied 5 or more hours?
  1. Stem-and-leaf display:
Stem | Leaves
   0 | 3 3 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 7 8
   1 | 2

  1. Shape: Roughly symmetric with a possible outlier; unimodal Center: Around 5 hours Spread: 3 to 12 hours (range of 9) Outliers: 12 hours appears to be an outlier

  2. Count of values ≥ 5: 13 out of 20 Relative frequency = 13/20 = 0.65 or 65%

Before You Leave

Today

  • Stem-and-leaf displays: preserve data while showing shape
  • Dotplots: simple visual for small datasets
  • Histograms: frequency distributions for larger datasets
  • Describing distributions: Shape, Center, Spread, Outliers (S-C-S-O)

Any questions?

Next Lesson

Lesson 3: Measures of Location

  • Mean, median, and mode
  • Percentiles and quartiles
  • Comparing measures of center for different distributions

Upcoming Graded Events

  • WebAssign 1.3 - Due before Lesson 3
  • Exploratory Data Analysis - Due Lesson 9
  • WPR I - Lesson 16