Measures of Center

🔍 Exploratory Data Analysis

Part 1 of 7 — EDA Overview

What Is EDA?

Exploratory Data Analysis (EDA) is the process of using graphs and summary statistics to understand the key features of a dataset.

The Four Features (SOCS)

When describing a distribution, always mention:

Types of Data

Graphs for Categorical vs. Quantitative

• Categorical: bar chart, pie chart
• Quantitative: histogram, stemplot, boxplot, dotplot

Concept Check 🎯

Data Classification 🧮

Classify each as categorical (C) or quantitative (Q):

1) Zip code

2) Temperature in degrees Fahrenheit

3) Number of siblings

📊 Graphical Displays

Part 2 of 7 — Graphs for Quantitative Data

Histograms

• Bars represent frequency (or relative frequency) for intervals
• No gaps between bars (unlike bar charts)
• Show shape, center, spread, and outliers

Stemplots (Stem-and-Leaf Plots)

• Each value is split into a “stem” and “leaf”
• Good for small datasets (preserves individual values)
• Back-to-back stemplots compare two groups

Dotplots

• Each value represented by a dot above a number line
• Best for small datasets
• Easy to see clusters, gaps, and outliers

Comparative Displays

To compare distributions, use:

• Side-by-side boxplots
• Back-to-back stemplots
• Overlapping or stacked histograms

🔑 Always compare shape, outliers, center, AND spread when comparing distributions.

Concept Check 🎯

Graph Selection 🧮

Choose the best graph type for each:

1) Comparing test score distributions of two classes (histogram/boxplot/stemplot)

2) Showing individual values of 20 measurements (histogram/dotplot/boxplot)

3) Displaying the distribution of 500 exam scores (histogram/dotplot/stemplot)

📍 Measures of Center

Part 3 of 7 — Mean and Median

Mean ($\bar{x}$)

$\bar{x} = \frac{\sum x_i}{n}$

• Arithmetic average of ALL values
• Sensitive to outliers and skewness
• Best for symmetric distributions

Median ($M$ or $\tilde{x}$)

• Middle value when data is ordered
• Resistant to outliers
• Best for skewed distributions

Relationship Between Mean and Median

Why Does Skewness Pull the Mean?

Outliers and long tails pull the mean toward the tail, while the median stays put.

Concept Check 🎯

Center Calculations 🧮

Data: 10, 12, 14, 15, 19

1) Mean $\bar{x} = ?$

2) Median = ?

3) If we add 100 to the dataset, which changes more — the mean or the median?

📏 Measures of Spread

Part 4 of 7 — Range, IQR, Standard Deviation

Range

$\text{Range} = \max - \min$

Simple but not resistant to outliers.

Interquartile Range (IQR)

$IQR = Q_3 - Q_1$

• Middle 50% of the data
• Resistant to outliers
• Best for skewed distributions

Standard Deviation ($s$)

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

• Measures a typical distance of the values from the mean
• NOT resistant to outliers
• Best for symmetric distributions
• $s = 0$ only when all values are identical

Outlier Rule (1.5 × IQR)

A value is an outlier if it falls:

• Below $Q_1 - 1.5 \times IQR$
• Above $Q_3 + 1.5 \times IQR$

Concept Check 🎯

Spread Calculations 🧮

Five-number summary: Min = 5, $Q_1 = 12$, Median = 18, $Q_3 = 24$, Max = 50.

1) $IQR = ?$

2) Upper outlier fence = $Q_3 + 1.5 \times IQR = ?$

3) Is the maximum value (50) an outlier? (yes/no)

📈 Shape and Outliers

Part 5 of 7 — Describing Distributions Completely

Shapes of Distributions

The Effect of Outliers

Outliers affect:

• Mean (pulled toward outlier) — NOT resistant
• Standard deviation (increases) — NOT resistant
• Range (increases) — NOT resistant

Outliers do NOT significantly affect:

• Median — resistant
• IQR — resistant

Boxplots

The five-number summary is displayed visually: $\text{Min}, Q_1, \text{Median}, Q_3, \text{Max}$

Modified boxplots show outliers as individual dots beyond the fences.

Concept Check 🎯

Distribution Description 🧮

A histogram of household income in a city shows a peak around $50,000 with a long tail stretching past $500,000.

1) What is the shape? (symmetric/skewed right/skewed left)

2) Is the mean likely above or below the median?

3) Which is a better measure of center for this data? (mean/median)

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

AP Exam Framework for EDA

When describing a distribution, ALWAYS address:

1. Shape (is it symmetric? skewed? bimodal?)
2. Outliers (are there any? use the 1.5×IQR rule)
3. Center (give an approximate value and name the statistic)
4. Spread (report the appropriate measure)

Template Answer

“The distribution of [variable] is [shape] with [center measure] approximately [value] and [spread measure] approximately [value]. [There are / are no] outliers.”

Comparing Distributions

Always compare both distributions on ALL four features. Use comparative language: “higher,” “wider,” “more skewed.”

Concept Check 🎯

SOCS Description 🧮

Test scores: Min=45, $Q_1=65$, Med=75, $Q_3=85$, Max=98. Roughly symmetric, no outliers.

1) Best measure of center? (mean/median) and its approximate value?

2) $IQR = ?$

3) Lower fence for outliers by the 1.5×IQR rule = ?

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Quick Reference

EDA Checklist

• Identify variable type (categorical vs. quantitative)
• Choose appropriate graph
• Describe shape (symmetric, skewed L/R, bimodal, uniform)
• Check for outliers (1.5 × IQR rule)
• Report center (mean or median)
• Report spread (SD or IQR)
• Use context (variable names, units)

Concept Check 🎯

Final Challenge 🧮

Data: 3, 5, 7, 8, 9, 10, 12, 14, 50

1) $\bar{x} = ?$ (round to 1 place)

2) Median = ?

3) Is 50 an outlier by the 1.5×IQR rule? ($Q_1 = 6$, $Q_3 = 13$, so $IQR = ?$ and upper fence = ?)