Dotplot

Structure:

• Horizontal axis: values of variable
• Dots stacked vertically for each value
• One dot = one data point

Advantages: sees every individual value, shows gaps and clusters Disadvantages: crowded with large datasets

Stemplot (Stem-and-Leaf Plot)

Structure:

• Stem: tens digit (left side)
• Leaf: ones digit (right side)
• Leaves ordered left-to-right

Example: Dataset 12, 15, 18, 21, 23, 25

1 | 2 5 8
2 | 1 3 5

Interpreting: stem = 1, leaf = 2 means 12

Back-to-back stemplot: compare two distributions

Group A | stem | Group B
  8 5 2 |  1   | 3 4 7
    1 0 |  2   | 2 5 8

Histogram

Structure:

• Bins (class intervals) on x-axis
• Frequency (count) on y-axis
• Bars touch (data is continuous)
• Height = frequency (or relative frequency ÷ width)

Key choice: width of bins

• Too wide: lose detail
• Too narrow: too fragmented

Important: area of bar = relative frequency when using density scale

Boxplot

Structure:

• Box: from Q1 to Q3 (middle 50%)
• Line in box: median (Q2)
• Whiskers: extend to minimum/maximum (or 1.5·IQR rule)
• Dots: outliers beyond whiskers

Formula for outlier detection:

• Lower fence: \(Q1 - 1.5(IQR)\)
• Upper fence: \(Q3 + 1.5(IQR)\)
• Points outside fences are outliers

Ogive (Cumulative Distribution)

Structure:

• x-axis: values
• y-axis: cumulative relative frequency (0 to 1 or 0% to 100%)
• Points connected by line segments
• Always increasing (non-decreasing)

Use: find percentiles

• Read up from x-value to curve, then left to y-axis
• Or read left from y-axis to curve, then down to x-axis

Worked Example

Data: Test scores for 20 students: 62, 68, 71, 74, 74, 75, 77, 78, 80, 81, 82, 83, 85, 86, 87, 88, 90, 92, 94, 98

Stemplot:

6 | 2 8
7 | 1 4 4 5 7 8
8 | 0 1 2 3 5 6 7 8
9 | 0 2 4 8

Histogram (bins 60-70, 70-80, 80-90, 90-100):

• [60, 70): 2 students
• [70, 80): 6 students
• [80, 90): 8 students
• [90, 100]: 4 students

Boxplot: Q1 ≈ 74.5, Q2 ≈ 82.5, Q3 ≈ 87.5, range 62–98, no outliers

Common Mistakes

1. Histogram bars not touching: bars should touch (continuous data)
2. Mislabeling x-axis in stemplot: leaves must be single digit
3. Ignoring class width in histogram: density = frequency ÷ width
4. Outlier calculation: use 1.5·IQR rule (not "looks far away")
5. Confusing relative vs. cumulative: ogive uses cumulative, histogram uses frequency

AP Exam Tip

When asked to "display" data, choose the method that best shows the feature in question:

• Comparing shapes of two distributions? → Parallel boxplots or back-to-back stemplots
• Seeing exact values? → Dotplot or stemplot
• Large dataset? → Histogram
• Finding a percentile? → Ogive

Always label axes and title your graph.

Wait, let me recalculate. Scores are: 8, 9, 9, 10, 10, 10, 11, 12, 12, 13

Better notation:

• 8 = stem 0, leaf 8 (or stem 8, leaf nothing)
• 9, 9 = stem 0, leaves 9, 9
• 10, 10, 10 = stem 1, leaves 0, 0, 0
• etc.

$\text{Stem} | \text{Leaf}$ $0 | 8 \; 9 \; 9$ $1 | 0 \; 0 \; 0 \; 1 \; 2 \; 2 \; 3$

Or in traditional format: $8 | \text{ }$ $9 | 0 \; 0$ $10 | 0 \; 0 \; 0$ $11 | 0$ $12 | 0 \; 0$ $13 | 0$

Interpretation: Most scores cluster at 10; shape is roughly symmetric with a slight left skew (tail toward lower scores). No outliers.