Three Ways to Display Numerical Data

In this lesson:

• Construct dot plots, histograms, and box plots
• Interpret what each display shows about a data set

Learning Objectives for Today's Lesson

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

1. Construct a dot plot by plotting each value on a number line
2. Construct a histogram using equal-width frequency bars
3. Construct a box plot from the five-number summary
4. Interpret each display: center, spread, shape, outliers

Same Data, Three Different Views

Each display reveals something the others don't

Dot Plots: Seeing Every Value

• Each dot represents exactly one data value
• Dots stack vertically when values repeat
• The number line shows the full range of the data

Best for: small data sets (fewer than ~30 values)

Building a Dot Plot: Four Steps

• Step 1: Draw a number line spanning the data range
• Step 2: Choose a scale so all values fit without crowding
• Step 3: Place one dot per data value at its position
• Step 4: Label the axis with the variable name and units

Dot Plot Example: Books Read This Summer

Data: Books read by 15 students:

Reading the dot plot:

• Center: dots cluster around 4–5 books
• Spread: values range from 2 to 8
• Shape: slightly right-skewed (one value at 8)

Quick Check: Reading a Dot Plot

A dot plot shows quiz scores for 12 students.
The stack at score 7 has 3 dots.

Questions:

1. How many students scored 7?
2. What does a taller stack mean?

Think before the next slide...

What Dot Plots Show Well

Dot plots reveal:

• Exact position of every data value
• Clusters, gaps, and outliers
• Overall shape of the distribution

Dot plots hide:

• Frequency patterns in large data sets
• Large data sets become cluttered

Transition: From Dots to Intervals

When data sets grow large, individual dots crowd the plot.

Histograms solve this by grouping values into equal-width intervals — so you see the shape, not every single value.

Histograms: Grouping Values into Intervals

• Data is divided into equal-width bins (intervals)
• The height of each bar = the frequency (count) in that bin
• Bars touch — no gaps between them

Best for: larger data sets (20+ values)

Building a Histogram: Five Steps

• Step 1: Determine the range of the data
• Step 2: Choose equal-width bins (aim for 4–8 bins)
• Step 3: Count values in each bin
• Step 4: Draw bars with height = count; bars must touch
• Step 5: Label x-axis with bin boundaries, y-axis with frequency

Histogram Example: Student Heights in Centimeters

Data: Heights (cm) of 20 students; bins of width 5 cm

Shape: unimodal, slightly right-skewed; center ~145–150 cm

Quick Check: Reading a Histogram

A histogram shows 30 students' quiz scores.

• The bar for 70–80 reaches a height of 12
• The bar for 80–90 reaches a height of 8

Questions:

1. How many students scored between 70 and 80?
2. Which interval has fewer students?

Try before advancing...

Histogram Bars Must Always Touch

Histogram vs. Bar Graph — a critical difference:

• Bar graph: gaps between bars → data is categorical
• Histogram: bars touch → data is numerical, continuous

A gap implies no values are possible between intervals — never true for numerical data.

Your Turn: Build a Histogram Now

Data (16 days, °F):

1. Use bin width 5°F
2. Count values per bin
3. Draw bars; label both axes

Try it, then check the next slide

Transition: From Shape to Summary

Histograms show shape well — but hide individual values.

Box plots give you a compact summary of the center and the spread of the middle half of any data set.

They're ideal for comparing two data sets side by side.

Box Plots Use a Five-Number Summary

IQR = Q3 − Q1 = width of the box

Anatomy of a Box Plot Diagram

The box spans Q1 to Q3 — it covers the middle 50% of the data

Computing Q1 and Q3 Step by Step

Sorted scores (11 values):

• Median = 75 (6th value of 11)
• Lower half: 45, 60, 65, 70, 72 → Q1 = 65
• Upper half: 75, 80, 85, 90, 95 → Q3 on next slide

Drawing the Box Plot from Summary

From previous: Min = 45, Q1 = 65, Median = 75

• Upper half: 75, 80, 85, 90, 95 → Q3 = 85; Max = 95

Five-number summary: 45 | 65 | 75 | 85 | 95

Box: Q1=65 to Q3=85; median at 75; whiskers to 45 and 95

Quick Check: Read This Box Plot

A box plot: whisker ends at 40 and 100; box edges at 55 and 80; median line at 65.

Questions:

1. What is Q1? What is Q3?
2. What is the IQR?
3. Is the median centered in the box?

Work it out before advancing...

Symmetry and Skewness in Box Plots

Reading shape from a box plot:

• Symmetric: median line centered in the box; whiskers equal length
• Left-skewed: median closer to Q3; left whisker longer
• Right-skewed: median closer to Q1; right whisker longer

The direction of the longer whisker shows the direction of the skew

Watch Out: Width Does Not Mean Count

Each of the four sections always contains exactly 25% of the data — regardless of visual width.

• Longer whisker → values are more spread out, not more numerous
• Wider box half → values are farther apart, not more of them

Strengths and Limits of Box Plots

Box plots reveal:

• Center (median) and spread (IQR = box width)
• Symmetry/skewness from median position and whisker lengths
• Great for comparing two data sets side by side

Box plots hide:

• Exact values; count of data points per region

Choosing the Right Display for Data

12 scores — show every individual value
→ Dot plot — small data, each value visible

200 heights — show which range appears most
→ Histogram — large data, shape and frequency

Two classes — compare center and spread
→ Box plot — compact summary, side-by-side

Your Turn: Choose the Best Display

Decide: dot plot, histogram, or box plot?

1. 18 data values — show each one on a number line.
2. 60 values — show which range appears most often.
3. Compare median test scores for two classes.

Write your answers — check the next slide

Answers: Which Display Fits Best?

1. 18 values, show each → Dot plot ✓
   — small data; every value visible on a number line

2. 60 values, most common range → Histogram ✓
   — large data; frequency by interval

3. Compare two classes → Box plot ✓
   — compact five-number summary; side-by-side

Summary: Three Data Display Tools

Dot plot: one dot per value; stacked = repeat; small data

Histogram: bars = frequency per bin; bars touch; large data

Box plot: box = IQR; median line inside; whiskers = range

Histogram bars must touch — no gaps

Q1 = median of lower half; exclude overall median

Wider section ≠ more data — every section = 25%

Coming Up Next: Quantitative Data Summaries

Next lesson — 6.SP.B.5:

• Summarize data sets quantitatively
• Calculate mean, median, and IQR
• Choose the best measure of center and variation

The displays from today are the visual context for those calculations.