Box Plots and Choosing the Right Plot

In this lesson:

• Compute the five-number summary from ordered data
• Construct and read a box plot
• Choose the right plot for the question being asked

What You Will Learn Today

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

1. Compute the five-number summary: min, Q1, median, Q3, max
2. Construct a box plot on a number line from the five-number summary
3. Read center, spread, and shape from a box plot
4. State the 25%-per-region rule and apply it to a given box plot
5. Choose the most informative plot type for a given data size and question

Same Heights — Two New Questions

Which value is the exact middle of the 25 heights?

Where does the "middle half" of the data sit?

52, 54, 54, 55, 55, 55, 56, 56, 56, 56, 57, 57, 57, 58, 58, 58, 59, 59, 60, 60, 61, 62, 63, 63, 65

Why Mark the Median Permanently?

52, 54, 54, 55, 55, 55, 56, 56, 56, 56, 57, 57, 57, 58, 58, 58, ...

The median is the 13th value = 57 inches

Counting works for 25 values — but imagine 250. The box plot marks it permanently.

Five Landmarks in Every Data Set

Every data set has five key anchors:

• Minimum — smallest value
• Q1 — median of the lower half (25th percentile)
• Median — middle value (50th percentile)
• Q3 — median of the upper half (75th percentile)
• Maximum — largest value

Step 1: Find the Median

52, 54, 54, 55, 55, 55, 56, 56, 56, 56, 57, 57, 57, 58, 58, 58, 59, 59, 60, 60, 61, 62, 63, 63, 65

• 12 values below · 12 values above
• Median (57) excluded from both halves

Step 2: Find Q1 from the Lower Half

Lower half — 12 values:
52, 54, 54, 55, 55, 55, 56, 56, 56, 56, 57, 57

• 6th value = 55, 7th value = 56
• Q1 = 55.5 inches

Step 3: Find Q3 from the Upper Half

Upper half — 12 values:
58, 58, 58, 59, 59, 60, 60, 61, 62, 63, 63, 65

• 6th value = 60, 7th value = 60 → Q3 = 60 inches

Five-number summary: 52, 55.5, 57, 60, 65

The Odd-n Quartile Convention Explained

When n is odd, the median sits alone in the center — excluded from both halves.

• Lower half: first 12 values (not 13)
• Upper half: last 12 values (not 13)

This course always uses the exclude-median convention.

Five Numbers Become One Picture

Five landmarks → one picture: whisker–box–line–box–whisker

Four Regions, Each Holding 25%

• Left whisker (52 → 55.5): bottom 25%
• Left half of box (55.5 → 57): next 25%
• Right half of box (57 → 60): next 25%
• Right whisker (60 → 65): top 25%

Length describes spread — not how many values are there.

Box Width Equals the IQR

• IQR = spread of the middle 50%
• Median (57) slightly left of center → right skew
• Right whisker longer (5 in.) than left (3.5 in.) → confirms skew

Read This Box Plot: Four Questions

Min = 40, Q1 = 48, Median = 53, Q3 = 58, Max = 70

1. What is the IQR?
2. What is the range?
3. Which whisker is longer?
4. Left-skewed, right-skewed, or symmetric?

Answer all four before advancing.

Longer Whisker — Does It Hold More Data?

A box plot's right whisker is twice as long as the left.

• A. Yes — longer means more data there
• B. No — each whisker always holds 25%

Commit to A or B before advancing.

Every Region Holds 25% — Length ≠ Count

• Left whisker (3.5 in., 52→55.5): 6–7 students — same as right
• Right whisker (5 in., 60→65): 6–7 students — more spread, not more

Long whisker = outer quarter is spread out, not more numerous.

Each Plot Answers a Different Question

• Dot plot — answers "what are the individual values?"
• Histogram — answers "what does the shape look like?"
• Box plot — answers "where is the middle, and how do groups compare?"

Decision Rule: Which Plot Fits the Question?

• Small n (≤ ~30), individual values matter → dot plot
• Large n, shape of distribution matters → histogram
• Comparing two or more groups → side-by-side box plots

Two Classes — Side by Side

Full Practice: Compare and Choose

1. Higher median?
2. Larger IQR?
3. "Higher typical values" — best plot?
4. "Show P's shape to large audience" — best plot?

Which Plot Fits Each Scenario?

1. Teacher: 28 quiz scores — report each student's exact score
2. Journalist: 3,000 income values — show distribution shape
3. Coach: sprint times for three teams — compare the groups

Name the plot type for each. Give one reason.

Box Plots and Choosing Plots: Key Ideas

✓ Five-number summary: min, Q1, median, Q3, max

✓ Box Q1→Q3; median line inside; whiskers to min/max

✓ IQR = box width; each region = 25% — length ≠ count

✓ Dot → values; histogram → shape; box plot → groups

Odd n: exclude median when computing Q1/Q3

What's Next: Summarizing Data in 6.SP.B.5

In 6.SP.B.5, you'll write complete data summaries:

• Measure of center (mean or median)
• Measure of variation (IQR or MAD)
• Context: attribute, units, group