Box Plots and Choosing the Right Plot

Lesson 2 of 2: Represent Data with Plots

In this lesson:

• Construct box plots using the five-number summary
• Interpret outliers and distribution shape from box plots
• Choose the best display for any data situation

Learning Objectives

By the end of this unit, you should be able to:

1. Construct dot plots, histograms, and box plots to represent data
2. Choose the most appropriate plot type for the data and question
3. Interpret patterns, outliers, and distribution shape from each plot
4. Compare the strengths and limitations of each plot type
5. Use technology (spreadsheets, calculators) to create data displays efficiently

Where We Left Off

Dot plots (Lesson 1): Show every individual value — best for small data sets

Histograms (Lesson 1): Show distribution shape — best for large data sets

Today — Box plots: Summarize with 5 key numbers — best for comparing groups

Each tool has different strengths. By the end of this lesson, you'll know how to choose.

Box Plots: Five Numbers Tell the Story

A box plot (box-and-whisker plot) summarizes data using five key statistics:

Together these are the five-number summary

Step 1: Sort the Data

Ages of 19 people at a community event (already sorted):

12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 28, 30, 32, 35, 40, 45, 50

• Minimum: 12 (lowest value)
• Maximum: 50 (highest value)
• Median: 10th of 19 values → 24

19 values: median is the 10th value in sorted order

Step 2: Find Q1 and Q3

Lower half (values below median 24): 12, 14, 15, 16, 17, 18, 20, 22, 23

Upper half (values above median 24): 25, 26, 28, 30, 32, 35, 40, 45, 50

Five-number summary: Min = 12 · Q1 = 17 · Median = 24 · Q3 = 32 · Max = 50

Step 3: Construct the Box Plot

• Box spans Q1 to Q3 → contains the middle 50% of the data
• Line at median divides the box in half
• Whiskers extend to min and max (the other 50%)

The box shows the middle 50% — not all the data

The Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of data.

Outlier fences (using the 1.5 × IQR rule):

Values outside [−5.5, 54.5] are outliers. Our data: all values between 12 and 50 → no outliers

What Happens with an Outlier?

Revised scenario: The oldest person is 80 (not 50)

• Upper fence = 54.5 → 80 > 54.5 → 80 is an outlier
• Upper whisker stops at 50 (largest non-outlier)
• Outlier plotted as a separate dot at 80

Watch out: An outlier is not automatically a mistake — it may be a real, unusual value

Always investigate: data entry error? Measurement error? Or a genuine rare event?

Reading Skewness from Box Plots

Box plot shape reveals distribution skewness:

• Right-skewed: Right whisker longer, median closer to Q1
  • Our data: right whisker (32 to 50) is longer → right skew
• Left-skewed: Left whisker longer, median closer to Q3
• Symmetric: Both whiskers similar, median near center of box

Watch out: An off-center median line is not an error — it tells you the data is skewed

Quick Check: Read a Box Plot

Suppose a box plot shows: min = 60, Q1 = 72, median = 78, Q3 = 85, max = 98

1. What is the IQR?
2. What are the outlier fences? Are there any outliers?
3. Where is 50% of the data?
4. Describe the shape: symmetric, right-skewed, or left-skewed?

Answers: IQR = 85 − 72 = 13 · Lower fence = 72 − 19.5 = 52.5; Upper fence = 85 + 19.5 = 104.5; no outliers · 50% of data between 72 and 85 · Roughly symmetric (whiskers similar length)

Side-by-Side Box Plots: Comparing Groups

• Which class scored higher? → Class A (higher median)
• Which class had more variability? → Class B (larger IQR, wider box)
• Any outliers? → Check dots beyond whiskers

Box plots make group comparisons fast and visual

Your Turn: Build a Box Plot

Class A test scores (already sorted):
72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 94

Your tasks:

1. Find: minimum, Q1, median, Q3, maximum
2. Calculate IQR and outlier fences
3. Draw the box plot on a number line from 65 to 100
4. Describe the shape

Work through each step before comparing with your partner

Your Turn: Side-by-Side Comparison

Class B test scores (already sorted):
60, 68, 72, 74, 78, 82, 86, 90, 94, 96, 98

Your tasks:

1. Find the five-number summary for Class B
2. Draw Class B's box plot below Class A's, on the same scale
3. Write a comparison: "Class A has a median of ___ versus Class B's median of ___. Class ___ shows greater variability (IQR = ___). There are/are no outliers in either class."

Compare on the same scale — that's what makes the visual comparison meaningful

Three Tools, Three Strengths

When choosing a data display, ask two questions:

1. How large is your data set? (Small → dot plot; Large → histogram or box plot)
2. What question are you answering? (Individual values? Shape? Comparison?)

Same Data, Three Displays

Texts per day for 25 teenagers

Each display reveals different features — and hides different features

What Each Plot Reveals

Scenario 1: Penguin Weights

Situation: 200 adult penguins from 3 species; researcher wants to compare weights and identify unusual individuals

• Dot plot? No — 200 values per species → too cluttered
• Histogram? Possible, but comparing 3 histograms side by side is harder
• Box plots? ✓ Best choice

Reason: Three side-by-side box plots allow instant comparison of medians, spreads, and outliers across all three species

Scenario 2: Quiz Scores for 12 Students

Situation: 12 quiz scores — teacher wants to show every score to the class

Data: 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10

• Dot plot? ✓ Best choice — 12 values is small; every score visible
• Histogram? Probably not — only 12 values; grouping hides small-sample detail
• Box plot? Could work, but loses the key insight that 6 students scored 10

Reason: Small data set where individual values carry meaning → dot plot wins

Scenario 3: 200 Tree Heights

Situation: 200 height measurements; scientist wants to know if the distribution is bell-shaped

• Dot plot? No — 200 values is too many
• Box plot? No — can't see distribution shape from a box plot
• Histogram? ✓ Best choice

Reason: Large data set with a shape question → histogram reveals shape directly

Decision Game: Your Turn

For each scenario, choose the best display and give a one-sentence reason:

A: 50 students' quiz scores (0–10). Want to see distribution shape.

B: 8 athletes' 100m sprint times. Coach wants to show every individual time.

C: 200 plants' heights measured in 3 different soil types. Compare the groups.

Write your answers before advancing

Answers: A → Histogram (large data set, shape question) · B → Dot plot (small data, individual values matter) · C → Box plots (multiple groups, comparison)

Decision Flowchart

Use this as a starting point — good answers depend on data size and question

Independent Practice

Choose the best plot type for each scenario. Write your choice and a one-sentence justification.

1. A teacher has quiz scores for 100 students and wants to identify any unusually low scores.
2. A health researcher has resting heart rates for 15 adult volunteers and wants to show every individual measurement.
3. A company has salary data for 500 employees across 4 departments and wants to compare distributions.
4. A meteorologist has 90 daily temperature readings and wants to know if the distribution is symmetric.
5. A coach has the vertical jump heights for 10 athletes and wants to find the median and compare to last year's team.

Choose and justify each — then compare with your partner

Key Takeaways

✓ Box plots summarize data with 5 numbers — best for comparing groups

✓ IQR rule: values beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR are outliers

✓ Choose by data size + question: dot plot (small, individual values) · histogram (large, shape) · box plot (groups, comparison)

Watch out: Box = middle 50% of data, not all data; 25% lies below the box

Watch out: Off-center median line means skewed data — not an error in the plot

Watch out: Outliers may be rare but real — investigate before removing

Next Lesson

HSS.ID.A.2 — Compare Center and Spread

• Use box plots to compare median and IQR across groups
• Use histograms to compare distribution shapes
• Discuss what makes distributions different: center, spread, shape, outliers

The displays you've built today are the tools for all future data comparison