📊 Graphical Representation of Data: Making Sense of Numbers

Welcome to the chapter on displaying data! Don't worry if Statistics sometimes feels overwhelming—this topic is all about turning raw tables and lists of numbers into clear, easy-to-read visual representations. When we visualize data correctly, patterns jump right out at us, making analysis much simpler.
In this section, we will learn how to choose the right chart for different types of data, how to construct them accurately, and how to interpret the information presented. Let’s get started!

1. Charts for Discrete and Categorical Data

Discrete data can only take specific, distinct values (like shoe size or count of goals scored), whereas categorical data describes qualitative labels (like favourite colour or mode of transport).

1.1 Pictograms

A pictogram uses picture symbols or icons to represent data frequencies.

Key Rules:

  • Every pictogram must include a key explaining what one whole symbol represents (e.g., Key: 1 circle = 4 books).
  • Partial frequencies are represented by dividing the symbol proportionally (e.g., half a circle for 2 books, a quarter circle for 1 book).
  • All symbols must be drawn the same size and spaced evenly in rows or columns.

1.2 Bar Charts and Vertical Line Graphs

A Bar Chart uses rectangular bars to show the frequency of different categories or discrete values.

Key Features:

  • The height of the bar represents the frequency.
  • The bars are drawn with equal widths and separated by gaps to show that categories or discrete values are distinct.
  • The horizontal axis is labelled with categories or discrete values, and the vertical axis is labelled with frequency.

Tip for Students:

If the data is numerical and discrete (such as test scores out of 10), a Vertical Line Graph can also be used, where a thin line represents each frequency instead of a bar.

1.3 Two-Way Tables

A two-way table organises data involving two different categorical variables simultaneously (for example, gender and chosen subject).

  • Rows display the categories of one variable; columns display the other.
  • Row totals and column totals must add up to the grand total in the bottom-right cell.
  • Missing values can be found by subtracting known values from the corresponding row or column total.

1.4 Pie Charts

Pie charts show how a total amount is divided into proportions or fractions of a whole.

Step-by-Step: Creating a Pie Chart

  1. Find the Total Frequency ( \(N\)).
  2. Calculate the sector angle for each category using the formula: \(\text{Angle} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ\)
  3. Draw a circle using compasses and measure each angle accurately with a protractor starting from the radius.
  4. Label each sector with the category name or include a key.

Common Mistake to Avoid: Always check that your calculated sector angles sum to exactly \(360^\circ\).

2. Dealing with Continuous Data (Grouped Frequency)

Continuous data is data that can take any value within a range (such as height, mass, or time). Continuous data is grouped into class intervals.

2.1 Histograms (Higher Tier Only)

Histograms are used to display continuous data, particularly when class intervals are of unequal widths. They look similar to bar charts, but operate on a fundamentally different principle.

Crucial Difference: Area vs. Height
In a standard bar chart, the height represents the frequency. In a Histogram, the Area of the bar is proportional to the Frequency:
\(\text{Area} = \text{Frequency}\)

Because the class widths vary, the vertical axis represents Frequency Density.

The Histogram Formula:
\(\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}\)

Step-by-Step: Drawing and Interpreting a Histogram
  1. Calculate the Class Width for each interval (\(\text{Upper Boundary} - \text{Lower Boundary}\)).
  2. Calculate the Frequency Density for each interval using \(\frac{\text{Frequency}}{\text{Class Width}}\).
  3. Plot the continuous variable on the horizontal axis and Frequency Density on the vertical axis.
  4. Draw bars with no gaps between them because continuous data has no breaks between intervals.

3. Cumulative Frequency Diagrams (Higher Tier Only)

Cumulative Frequency (CF) is the running total of frequencies up to the upper boundary of each class interval. It tells you how many values are less than or equal to a given value.

3.1 Constructing a Cumulative Frequency Diagram

  1. Calculate the cumulative frequency column by adding frequencies sequentially.
  2. Plot points at coordinates (\(\text{Upper Class Boundary}, \text{Cumulative Frequency}\)).
  3. Plot the start of the curve at (\(\text{Lowest Value}, 0\)).
  4. Join the plotted points with a smooth S-shaped curve (or straight line segments).

3.2 Estimating Statistical Measures

Let \(n\) be the total frequency. Using the vertical (cumulative frequency) axis:

  • Median: Draw a horizontal line across at \(\frac{n}{2}\) (or \(0.5n\)) to the curve, then read down to the horizontal axis.
  • Lower Quartile (\(Q_1\)): Read across at \(\frac{n}{4}\) (or \(0.25n\)) to the curve, then read down.
  • Upper Quartile (\(Q_3\)): Read across at \(\frac{3n}{4}\) (or \(0.75n\)) to the curve, then read down.

Interquartile Range (IQR):
\(\text{IQR} = Q_3 - Q_1\)
The interquartile range measures the spread of the middle 50% of the data and is not affected by extreme outliers.