Frequency Tables and Stem-and-Leaf Diagrams

Count each observation once

A frequency is the number of times a value or category occurs. List categories clearly, tally each observation into one category, then check that the frequencies add to the number of observations.

Worked example — organise the number of books borrowed

Twelve students borrow: 0, 1, 2, 1, 3, 2, 0, 2, 1, 4, 2, 2. Count repeats rather than adding the book numbers.

Here f means frequency. The five students in the “2 books” row count as five observations, not ten. Multiplying value by frequency will be useful later when finding a mean.

Use clear, non-overlapping class intervals

When there are many different values, group them into classes. Every possible observation should have one place. Do not leave a gap or put a boundary in two groups.

Worked example — place a boundary value

A time of exactly 10 minutes belongs in 10 ≤ t < 20, not the first class. A time of 39.8 goes in 20 ≤ t < 40. The final class includes 60 because the dataset has no time above 60.

The symbol ≤ includes the endpoint; < excludes it. These class widths are 10, 10, 20 and 20 minutes. For whole-number counts, classes such as 0–4 and 5–9 cover all allowed integers, but they would leave a gap for continuous values such as 4.5.

Grouping loses exact values. You know 16 times lie between 20 and 40 minutes, but not where each one lies. That is why many summaries from grouped data are estimates.

Classify in two ways with a two-way table

A two-way table records two classifications of the same observations. Row and column totals describe the same overall group; do not count them as extra people.

Worked example — recover and check a missing cell

If the Year 11 bus count is missing, use its row: 30 − 14 = 16. Check its column: 28 − 12 = 16. Both methods agree.

Compare proportions if the groups are different sizes. Here each year has 30 students: Year 10 has 18/30 = 60% walking, compared with 14/30 ≈ 46.7% in Year 11.

This lesson organises and compares the data. For selecting people and calculating event probabilities, use Sample Spaces and Two-Way Tables.

Construct an ordered stem-and-leaf diagram

Split each number into a stem and a final digit called a leaf. Sort the leaves on each row, retain repeats, and give a key with units. Keep an empty stem row when it lies between stems you use.

Worked example — organise recorded journey times

Times in minutes: 12, 18, 21, 14, 25, 12, 32, 29, 21. The tens digits are stems. Sort the units digits from the stem outwards.

There are nine leaves, matching nine observations. The smallest value is 12, the largest is 32. A leaf is one digit: 25 is placed as stem 2 and leaf 5, not stem 2 and leaf 25.

Worked example — use a decimal key

For 1.2, 1.2, 1.8 and 2.1 metres, stems 1 and 2 with leaves 2 2 8 and 1 work if the key states 1 | 2 means 1.2 m. The same-looking diagram can represent different values if its key changes.

Compare two sets back-to-back

Use shared stems in the centre. On the right, leaves increase from left to right. On the left, the smallest leaf sits nearest the stem, so the printed row decreases from left to right. Give a key for both sides.

Group A is the nine journey times above. Group B is 11, 15, 19, 20, 22, 24, 28, 31, 35. Both have nine observations, making a direct comparison easier. The middle values are 21 and 22 minutes respectively; B has the greater typical time by the median. The next averages lesson explains how to calculate and justify such comparisons.

A stem-and-leaf diagram is useful for a small or moderate numerical dataset because it preserves individual values. A very large dataset may be clearer in a grouped table or graph.

Think Like a Mathematician

A frequency table and a stem-and-leaf diagram can represent the same dataset. Grouping into wide intervals, however, discards information that cannot be reconstructed exactly from the grouped totals alone.

Common Mistakes

Counting totals twice: margins are checks, not extra observations.

Overlapping classes: decide which class owns each boundary.

Losing repeats: repeated data need repeated leaves.

Missing key: state the value and units on each side.

Practice

1. For 1, 0, 2, 2, 1, 3, 2, 0, give frequencies for 0, 1, 2 and 3.

2. For 0 ≤ x < 5, 5 ≤ x < 10 and 10 ≤ x < 20, where do 5 and 9.9 go?

3. Why are 0–5 and 5–10 poor class labels without further explanation?

4. A row total is 35 and its two cells are 19 and a missing value. Find the missing value.

5. Construct an ordered stem-and-leaf diagram for 23, 17, 21, 17, 30, 25.

6. A key says 4 | 6 means 4.6 kg. What does 5 | 2 mean?

7. In the back-to-back example, how many B times are at least 20 minutes?

8. Can you recover the exact 16 times in the 20 ≤ t < 40 class from its frequency alone?

Frequently Asked Questions

Must a stem-and-leaf diagram be ordered?

For this course, yes: sort leaves, keep repeats and supply a key. Pearson expects ordered diagrams even when the word ordered is omitted.

Are two-way tables only for probability?

No. They organise and compare two classifications. Probability uses their counts for a different purpose, which has its own lesson.

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