Stem and Leaf Plots

Definition & Purpose

  • Stem-and-Leaf Plot: A graphical technique for continuous or discrete quantitative data in which every observation is split into two parts

    • Stem: all digits except the final digit

    • Leaf: the final (right-most) digit

  • Main goal: display the frequency distribution while still preserving every raw data value

  • Acts as a midway point between a simple ordered list and a histogram

    • Keeps numerical precision (each data point remains visible)

    • Also conveys overall shape, spread, and possible clusters or gaps

Construction Procedure (Step-by-Step)

  1. Gather data

    • Example data set of IQ scores (5 observations): 89,  96,  105,  109,  118{89,\;96,\;105,\;109,\;118}

  2. Separate into stems & leaves

    • 89 → stem 88, leaf 99

    • 96 → stem 99, leaf 66

    • 105 → stem 1010, leaf 55

    • 109 → stem 1010, leaf 99

    • 118 → stem 1111, leaf 88

  3. Draw the framework

    • Draw a long vertical line down the page

    • Left of the line = stems (ascending order, one per row)

    • Right of the line = leaves (individual digits, listed horizontally for each stem)

  4. List stems once, in ascending order

    • For the example, stems are 8,9,10,118, 9, 10, 11

  5. Enter each leaf next to its matching stem

    • 8 | 9 (represents 89)

    • 9 | 6 (represents 96)

    • 10 | 5 9 (represents 105, 109)

    • 11 | 8 (represents 118)

  6. (Optional) Order leaves: Leaves can be sorted to improve readability; e.g. 10  ∣  5  910 \;|\;5\;9 rather than 9  59\;5

  7. Add a key or legend so readers know how to read numbers (e.g. "8 | 9 = 89")

Finished Example

Stem | Leaf
------------
  8  | 9
  9  | 6
 10  | 5 9
 11  | 8
Key: 8 | 9 represents 89

Interpretation Techniques

  • Frequency: Count leaves for any stem → shows how many observations fall in each stem interval (analogous to bar height in a histogram)

  • Range: Range=Max−Min=118−89=29\text{Range} = \text{Max} - \text{Min} = 118 - 89 = 29

  • Central tendency & spread: By “tilting your head” right, the leaves resemble the bars of a histogram, letting you infer skewness, modality, and clustering

  • Outliers: Unusually long gaps between stems or isolated leaves flag potential outlying values

Comparison to Histograms

Feature

Stem-and-Leaf

Histogram

Raw data visible?

Yes (exact digits)

No (grouped into bins)

Visual impact

Moderate

High

Quick shape recognition

Good, but must mentally rotate

Very good

Typical use cases

Exploratory data analysis with small-to-moderate nn (≈ n≤200n \le 200)

Any nn, especially large

Key takeaway: A stem-and-leaf plot is “a histogram with the numbers still attached.”

Ethical / Philosophical Angle

  • Transparency: Because every data point is shown, the plot embodies an ethic of full disclosure—no aggregation hides individual values

  • Pedagogical clarity: Bridges the conceptual gap between raw data tables and abstract frequency graphs, aiding learners’ understanding of distributional thinking

Recap Checklist for Exams

  • Define stem & leaf clearly

  • Reproduce the 5-step construction process from memory

  • Be able to convert a small data set to a stem-and-leaf plot quickly

  • Know how to read frequency, range, shape, and spot outliers

  • Articulate advantages vs. a histogram (detail vs. visual punch)

  • State typical use cases and limitations