Year 9 Maths The Thai Guy

Lesson 10.2 · 25 min

Back-to-Back Stem-and-Leaf Diagrams

Showing two data sets on one shared stem, reading off averages and ranges, and comparing the two.

In this lesson you will learn to…

  • Draw a back-to-back stem-and-leaf diagram with a key
  • Read the median, mode and range for each side
  • Compare two distributions using an average and the range
  • Make a sensible judgement from the comparison

Why this matters

When you want to compare two groups — two machines, two classes, two clubs — a back-to-back stem-and-leaf diagram puts them side by side while keeping every single value. You can see the shape of each data set and work out exact statistics from it.

Reading from the left

The left-hand side reads backwards from the stem. In the row 7  4  2  1  0∣57\;4\;2\;1\;0 \mid 5, the values are 50, 51, 52, 54 and 57 — the 0 nearest the stem is the smallest.

Which average?

StatisticTells youWatch out
Medianthe middle valuethe best general comparison
Modethe most common valuecan be misleading with small data sets
Meanthe balance pointaffected by extreme values
Rangehow spread out the data isdepends on the two most extreme values only

A good comparison sentence mentions a statistic and the context: “The median age in Club B (51) is much higher than in Club A (14), so Club B is probably the golf club.”

Key rules and formulas

  • One stem, two sides

    The stems go down the middle. One data set's leaves go to the right, the other's to the left. Leaves on both sides get closer to the stem as they get smaller.

  • Always give a key

    For example: mm on the right and mm on the left. Without a key, nobody knows whether means 35, 3.5 or 350.

  • Keep the data

    Unlike a grouped table, a stem-and-leaf diagram keeps every original value — so you can find the exact median, mode and range.

  • Comparing

    Compare an average (median or mean) to say which is generally bigger, and the range to say which is more consistent. Always relate it back to the context.

Worked examples

Worked example 1

Drawing the diagram

Fifteen students' results — Maths: 43, 63, 45, 57, 82, 52, 50, 69, 71, 51, 48, 70, 54, 90, 66. Geography: 42, 65, 75, 88, 73, 88, 57, 60, 72, 97, 79, 80, 62, 91, 51. Draw a back-to-back stem-and-leaf diagram.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Comparing

Use the diagram to decide which exam was easier.

Try it yourself first, then reveal the solution one step at a time.

Practice: back-to-back stem-and-leaf

9 questions

Nine questions about two machines that make washers. Key: mm (Machine B, right) and mm (Machine A, left).

  1. Question 1 of 9Complete the tableFoundation
    Read the extreme values from the washer diagram (in the intro above).
    Machine A (mm)Machine B (mm)
    Thickest
    Thinnest
    Range
  2. Question 2 of 9Type the answer
    Each machine made 25 washers. What is the median thickness for Machine A?
    mm
  3. Question 3 of 9Type the answer
    What is the median thickness for Machine B?
    mm
  4. Question 4 of 9Fill the gaps
    Give the modal thickness for each machine.

    Machine A: mm Machine B: mm

  5. Question 5 of 9Multiple choice
    Which machine produces thicker washers, on the whole?
  6. Question 6 of 9Type the answer
    Maths results for 15 students: 43, 63, 45, 57, 82, 52, 50, 69, 71, 51, 48, 70, 54, 90, 66. What is the median?
  7. Question 7 of 9Type the answer
    Geography results for the same students: 42, 65, 75, 88, 73, 88, 57, 60, 72, 97, 79, 80, 62, 91, 51. What is the mean?
  8. Question 8 of 9Choose all that apply
    Club A's ages have median 14 and range 32. Club B's have median 51 and range 47. One is a gymnastics club and one is a golf club. Select every sensible statement.

    Select every correct answer — there may be more than one.

  9. Question 9 of 9True or false
    A stem-and-leaf diagram without a key is still complete, because the numbers are all there.