Data Representation

Why Represent Data?

Raw data — a list of numbers or responses — is difficult to interpret. Data representation transforms raw data into charts, tables, and diagrams that reveal patterns, trends, and distributions at a glance.

Choosing the right chart for the type of data and the message you want to communicate is one of the most important skills in statistics. Using the wrong chart can obscure meaning or even mislead.

Choosing the Right Chart

Chart Type	Best For	Data Type
Bar chart	Comparing categories	Qualitative or discrete
Pie chart	Showing parts of a whole	Qualitative or discrete
Histogram	Showing distribution of grouped data	Continuous
Frequency polygon	Comparing distributions	Continuous
Stem-and-leaf	Showing spread, keeping raw values	Small discrete datasets
Line graph	Showing change over time	Continuous (time series)

Bar Charts

A bar chart uses rectangular bars of equal width to represent frequencies. The height (or length) of each bar corresponds to its frequency.

Key features of a good bar chart:

• Equal-width bars with gaps between them
• Both axes clearly labelled with units
• A descriptive title
• Bars start at zero on the frequency axis

Bar charts are for discrete or categorical data. For example, displaying favourite subjects or the number of goals scored per match.

A compound (grouped) bar chart places two or more bars side by side for each category, allowing direct comparison between groups.

Pie Charts

A pie chart divides a circle into sectors, where each sector represents a proportion of the total.

Calculating the angle for each sector:

• Angle = (frequency / total) × 360°

Worked example: 360 students were surveyed. 90 chose maths as their favourite subject.

• Angle = (90/360) × 360° = 90°

Pie charts are useful for showing proportions but difficult to compare between two datasets.

Histograms

A histogram looks like a bar chart but is used for continuous grouped data. Key differences:

• Bars have no gaps between them
• The area of each bar represents the frequency (not the height)
• The height of each bar is frequency density, not frequency

Frequency density = frequency ÷ class width

To find frequency from a histogram: frequency = frequency density × class width

Worked example: A bar has frequency density 4 and class width 5.

• Frequency = 4 × 5 = 20

Histograms are essential when class widths are unequal — bars of different widths must use frequency density to remain accurate.

Frequency Polygons

A frequency polygon is drawn by plotting points at the midpoint of each class interval at the corresponding frequency (or frequency density), then joining them with straight lines.

Steps:

1. Find the midpoint of each class interval.
2. Plot (midpoint, frequency) for each class.
3. Join the points with straight lines.
4. Connect to the x-axis at the start and end (or add half-classes of zero frequency).

Frequency polygons are useful for comparing two distributions on the same axes — something histograms cannot do clearly.

Stem-and-Leaf Diagrams

A stem-and-leaf diagram organises data by splitting each value into a stem (the leading digit(s)) and a leaf (the final digit).

Example data: 23, 27, 31, 34, 34, 38, 42, 45

Stem | Leaf
  2  | 3 7
  3  | 1 4 4 8
  4  | 2 5

Key: 2 | 3 means 23

Advantages: Shows the shape of the distribution while preserving all individual values. From an ordered stem-and-leaf, you can immediately read off the median and find the range.

A back-to-back stem-and-leaf diagram places two datasets either side of a shared stem, enabling direct comparison of two groups.

Frequency Tables

A frequency table tallies how many times each value (or class of values) occurs.

For grouped data, use class intervals that are equal in width (e.g. 10 ≤ x < 20, 20 ≤ x < 30).

Always use inequality notation to avoid ambiguity about which class a boundary value belongs to.

Example:

Score	Tally	Frequency
0-9	IIII	4
10-19	IIII II	7
20-29	III	3

Key Terms

Term	Meaning
Frequency	How many times a value occurs
Class interval	A range of values grouped together
Class width	The size of a class interval
Frequency density	Frequency ÷ class width (used in histograms)
Stem	The leading digit(s) in a stem-and-leaf diagram
Leaf	The final digit in a stem-and-leaf diagram

Common Mistakes

• Drawing a bar chart with touching bars — bars in a bar chart have gaps; only histograms have no gaps.
• Using height for frequency in histograms with unequal class widths — always use frequency density = frequency ÷ class width.
• Pie chart angles not summing to 360° — always check your angles add up to exactly 360° before drawing.
• Unordered stem-and-leaf diagrams — leaves must be written in ascending order to read the median correctly.
• Misreading the midpoint for a frequency polygon — use the midpoint of the class interval, not the lower or upper boundary.

Tips and Tricks

• For histograms, remember: area = frequency. If you need to find a frequency, calculate area (frequency density × class width).
• Use a protractor carefully for pie charts — small angle errors look large in the finished diagram.
• Before drawing any chart, check what type of data you have (discrete/continuous/categorical) to select the right chart type.
• An ordered stem-and-leaf diagram allows you to find the median in seconds — count to the middle value without sorting a long list.