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

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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.

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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, Median, Mode

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.

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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 average distance 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 $Q_3 + 1.5 \\times IQR$

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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.

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Distribution Description 🧮

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

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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) Are there outliers by the 1.5×IQR rule? (yes/no) Lower fence = ?

📝 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)

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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 = ?)