Representation of data – Representation of discrete data: stem-and-leaf diagrams

• Stem = leading digits (e.g. 16, 17, 18, 19 for heights in cm).
• Leaf = last digit (e.g. 4 means 164 cm when stem is 16).
• Leaves on each row are written in ascending order.
• Back-to-back diagram: one group on the left, the other on the right of the same stems.

The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
2. Find the median and the interquartile range of the heights of the players in the Aces.
3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.

Solution

(a) Back-to-back stem-and-leaf diagram:

Key: \( 1 \mid 17 \mid 0 \) means \(171\) cm (Aces) and \(170\) cm (Jets).

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(b) Using the diagram to find median and quartiles (Aces):

Aces ordered: 164, 166, 168, 169, 171, 173, 174, 180, 181, 182, 182.

• Median (6th value) \(= 173\ \text{cm}\).
• Lower quartile \(Q_1 = 168\ \text{cm}\).
• Upper quartile \(Q_3 = 181\ \text{cm}\).
• \(\text{IQR} = 181 - 168 = 13\ \text{cm}\).

(c) Comparing spreads (ranges):

• Aces range: \(182 - 164 = 18\ \text{cm}\).
• Jets range: \(190 - 166 = 24\ \text{cm}\).

Jets have the wider spread in heights.

• Always include a clear key for the stem-and-leaf diagram.
• Check that leaves on each row are in ascending order.
• Use the ordered data to find median, quartiles and IQR accurately.
• When comparing two groups, comment on both location (median) and spread (range / IQR).