Representation of Data

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Select a suitable way of presenting raw statistical data, and discuss advantages and/or disadvantages that particular representations may have
  2. Draw and interpret stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency graphs (including back-to-back stem-and-leaf diagrams)
  3. Understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation), for example in comparing and contrasting sets of data
  4. Use a cumulative frequency graph to estimate medians, quartiles, percentiles, the proportion of a distribution above (or below) a given value, or between two values
  5. Calculate and use the mean and standard deviation of a set of data (including grouped data) either from the data itself or from given totals Σx and Σx², or coded totals Σ(x - a) and Σ(x - a)², and use such totals in solving problems which may involve up to two data sets

1. Types of Data

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Before we can represent data, we need to understand what type of data we have.

Qualitative Data (also called categorical data)

This is data described by words, not numbers. Examples include:

  • Blood types (A, B, AB, O)
  • Colours (red, blue, green)
  • Types of pets (dog, cat, bird)

Quantitative Data

This is data that takes numerical values. Quantitative data splits into two types:

Discrete Data — Values that are counted and can only take certain specific values

  • Number of letters in a word (1, 2, 3, 4... but not 2.5)
  • Shoe sizes (5, 5.5, 6, 6.5... but not 5.73)
  • Number of students in a class (20, 21, 22... but not 20.4)

Continuous Data — Values that are measured and can take any value within a range

  • Time taken to run a race (could be 12.3 seconds, 12.34 seconds, 12.341 seconds...)
  • Height of a person (could be 165.2 cm, 165.23 cm...)
  • Temperature (could be 20.5°C, 20.52°C...)

Think of discrete data as dots on a number line and continuous data as a solid bar covering all possible values in a range.

2. Stem-and-Leaf Diagrams

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A stem-and-leaf diagram is a way to organize and display discrete data so you can see all the individual values while also seeing the overall pattern.

How to Construct a Stem-and-Leaf Diagram

Step 1: Split each number into a "stem" (all digits except the last one) and a "leaf" (the last digit)

  • For the number 58, the stem is 5 and the leaf is 8
  • For the number 142, the stem is 14 and the leaf is 2

Step 2: Write the stems in a vertical column in order (smallest to largest)

Step 3: Write each leaf next to its stem, arranging the leaves in order from smallest to largest

Step 4: Always include a key to explain what your diagram means

Example

Let's organize these test scores: 58, 55, 58, 61, 72, 79, 97, 67, 61, 77, 92, 64, 69, 62, 53

Solution:

StemLeaf
53 5 8 8
61 1 2 4 7 9
72 7 9
8
92 7

Key: 5 | 3 represents a score of 53%

Notice how stem 8 is empty — this is fine and should still be shown.

Back-to-Back Stem-and-Leaf Diagrams

These are used to compare two sets of related data using a single central stem.

Example: Comparing rainfall data for 2016 and 2017

2016Stem2017
9 8 5 1 001 2 2 3 4 6 7 8 8 9
7 6 3 2 1 011 3
02

Key: 5 | 0 | 6 represents 5 days in 2016 and 6 days in 2017

The leaves for 2016 go to the left of the stem (in reverse order, so they still increase as you move toward the stem). The leaves for 2017 go to the right of the stem (in normal order).

Advantages and Disadvantages

Advantages:

  • Shows all original data values
  • Easy to find the median and mode
  • Shows the shape of the distribution
  • Compact for small data sets

Disadvantages:

  • Becomes messy with large amounts of data
  • Not suitable for continuous data
  • Takes time to construct by hand

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