Box Plots and Choosing the Right Plot

Lesson 2 of 2: Represent Data with Plots

In this lesson:

• Build a box plot from the five-number summary
• Choose the right plot for the data and question

Learning Objectives for This Unit

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

1. Construct dot plots, histograms, and box plots
2. Choose the right plot for the data and question
3. Interpret patterns, outliers, and shape
4. Compare the strengths of each plot type
5. Use technology to build displays efficiently

Compare Four Classes on One Screen

You want to compare four classes' test scores at once.

• Four dot plots → a wall of dots
• Four histograms → crowded and clashing

We need each class squeezed into a compact summary we can line up.

Find the Median of the Ages

Sorted ages of 19 people at a community event:

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

With 19 values, the median is the 10th value: 24.

The Five-Number Summary of Data

Five numbers describe the whole data set:

• Minimum and Maximum — the two ends
• Median — the middle value
• Q1 and Q3 — the quartiles, cutting the data into quarters

A quartile is just a median — of half the data.

Find Q1 and Q3 from the Halves

• Q1 = median of the lower half = 17
• Q3 = median of the upper half = 32

Draw the Box and Whiskers

Box 17 to 32, median line at 24, whiskers to 12 and 50.

The Box Holds the Middle Half

The box does not hold all the data — it holds the middle 50%.

• 25% of values sit below the box
• 50% sit inside the box
• 25% sit above the box

The box is the typical middle, not the full range.

Predict: How Many Dots in the Box?

Of the 19 ages, how many fall inside the box? Predict, then count.

An Off-Center Median Means Skew

A median line off-center is information, not a mistake.

• Median near Q1 → upper half stretched → right skew
• Median near Q3 → lower half stretched → left skew
• Centered median → symmetric middle

Measure Spread with the IQR

The interquartile range measures the width of the box:

The IQR is the spread of the middle 50% — and ignores the extremes.

Set Outlier Fences from the IQR

A value is an outlier if it lies beyond a fence:

"1.5 IQRs past the box" is a principled meaning of far.

Predict: Change the Oldest Age to 80

Suppose the oldest age were 80 instead of 50.

• Which of the five numbers move?
• Does 80 become an outlier?

Predict each before the reveal.

Reveal: 80 Is an Outlier

The whisker stops at 45; 80 is plotted as a separate point.

An Outlier Is Unusual, Not Wrong

When you find an outlier, investigate — don't delete.

• It might be an error — a typo or a broken sensor
• It might be real — a 105° day in a heat wave

Check it, decide, and document. Don't silently erase data.

Your Turn: Build a Box Plot

Sorted house prices ($ thousands): 180, 200, 215, 230, 250, 270, 290, 310, 400

1. Find all five numbers
2. Draw the box, median line, and whiskers

Then describe the spread of the middle 50% in one sentence.

One Data Set Shown Three Ways

Daily texts for 25 teens — same data, three pictures, each hiding something.

Each Plot Reveals and Hides

No display does everything — what one reveals, another hides.

Two Questions Pick Your Plot

Before drawing anything, ask:

1. What do I want to see — values, shape, or a comparison?
2. How many data points do I have?

Small + every value → dot plot. Large + shape → histogram. Compare → box plots.

Choose a Plot: 200 Tree Heights

A scientist has 200 tree heights and asks: is the distribution bell-shaped?

• Want to see: the shape
• How many: 200 — too many for a dot plot

Shape + large set → histogram.

Choose a Plot: 8 Sprint Times

A coach has 8 sprint times and wants to show every athlete's time.

• Want to see: every individual value
• How many: only 8

Small set + individual values → dot plot.

Your Turn: Choose and Justify

A researcher compares test scores across four teaching methods, 50 students each.

• Which display fits — and why?

Choose a plot and justify it in one sentence before advancing.

Your Turn: Full Box Plot with Outlier Check

Sorted reaction times (ms): 180, 195, 200, 205, 210, 215, 220, 225, 230, 240, 320

1. Find the five-number summary
2. Compute the IQR and fences; test for an outlier
3. Draw the box plot

Three Box-Plot Traps to Avoid

The box holds the middle 50%, not all the data
An off-center median means skew, not a mistake
An outlier is unusual — investigate, don't delete

Two Questions, Three Different Displays

✓ Dot plot: every value in a small set
✓ Histogram: the shape of a large set
✓ Box plot: compare groups and flag outliers

Next: putting numbers on the center and spread these plots reveal.