Boxplots & Data Interpretation

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Read and interpret boxplots (minimum, Q1, median, Q3, maximum).
  • Compare center, spread, and skew using boxplots.
  • Understand IQR as a measure of spread.
  • Identify outliers conceptually.

Key Ideas

A boxplot shows:

  • Minimum
  • First quartile (Q1)
  • Median (Q2)
  • Third quartile (Q3)
  • Maximum

Interquartile Range (IQR):
\[ \text{IQR} = Q3 - Q1 \]

What Boxplots Reveal

  • Center: median
  • Spread: IQR (middle 50% of data)
  • Skew: left tail longer → left skew, right tail longer → right skew
  • Outliers: plotted as separate points (if given)

Common Problem Types

Reading Quartiles, Median, and Range

Identify the five-number summary directly from the boxplot.

Example:
Median is the line inside the box.

Comparing Centers (Medians)

Higher median → larger “typical” value.

Example:
Dataset A median = 40, Dataset B = 60
→ B has higher center.

Comparing Spreads (IQR and Range)

Longer boxes indicate more spread in the middle 50%.

Example:
Q1 = 20, Q3 = 50 → IQR = 30.

Identifying Skew From Whiskers

Longer whisker indicates direction of skew.

Example:
Right whisker longer → right-skewed.

Determining Overlap of Distributions

Check whether ranges overlap.

Example:
If A’s max < B’s Q1 → little or no overlap.

Spotting Outliers (When Shown)

Points beyond whiskers represent outliers.

Example:
A dot plotted away from the whisker → high outlier.

Strategies

  • Read quartiles left → right.
  • Focus on median for center.
  • Use IQR for spread, not just range.
  • Compare lengths of “whiskers” for skew clues.
  • Don’t infer exact frequencies from boxplots.

Worked Examples

Example 1 — Compare centers

Dataset A median = 50
Dataset B median = 60
→ Dataset B has higher center.

Example 2 — Which is more spread out?

If A’s IQR = 10 and B’s IQR = 25 → B is more spread out.

Example 3 — Identify skew

If right whisker is much longer → right-skewed.

Example 4 — Range

Range = max – min (from whiskers).
If min = 10, max = 70 → range = 60.

WarningCommon Mistakes
  • Confusing IQR with range.
  • Misreading skew direction (look at whiskers, not the box).
  • Thinking boxplot shows frequencies — it only shows quartiles.
  • Assuming all sections of a boxplot contain equal counts.

Practice Problems

  1. If Q1 = 20 and Q3 = 45, what is the IQR?
  2. A boxplot’s right whisker is longer. What kind of skew?
  3. Median of dataset = 32, median of another = 28. Which center is larger?
  4. min = 5, max = 30. What is the range?
  1. IQR = 45 − 20 = 25
  2. Right-skewed
  3. 32 is larger
  4. 30 − 5 = 25

Summary

  • Boxplots summarize center and spread using quartiles.
  • IQR measures spread of the middle 50%.
  • Whisker lengths help determine skew.
  • Look at medians first.
  • IQR reveals spread — not the whiskers alone.
  • Longer whisker → direction of skew.