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Frequency Tables and Diagrams

Organising and Displaying Data

Essential Understanding: Raw data is often difficult to understand. Frequency tables organise data by counting how often each value occurs, and diagrams give us a visual representation that makes patterns easy to spot.

Key Skill: Constructing frequency tables
CSEC Focus: Bar charts, pie charts, histograms

What is a Frequency Table?

Frequency

The frequency of a value is the number of times that value appears in a data set.

A frequency table lists each unique value (or class interval) alongside its frequency.

Symbol: Frequency is often denoted by \( f \), and the sum of all frequencies is \( \sum f = n \) (the total number of data points).

Constructing a Frequency Table with Tally Marks

Example: Students' Favourite Subjects

A class of 25 students were asked their favourite subject. The responses were:

Maths, English, Science, Maths, Art, Science, Maths, English, Maths, Science, Art, Maths, English, Science, Maths, Art, English, Science, Maths, Science, English, Maths, Art, Science, Maths

1
List each unique value (subject) in the first column
2
Go through the data and make a tally mark for each occurrence
3
Count the tally marks to get the frequency
Subject Tally Frequency (\(f\))
Maths |||| |||| 9
English |||| 5
Science |||| || 7
Art |||| 4
Total: \(\sum f = 25\)

Note: Tally marks are grouped in fives: |||| = 5. This makes counting easier!

Interactive Chart Builder

Data Visualizer

Enter frequency values below and watch the charts update in real-time!

Category Frequency
Apples
Bananas
Grapes
Oranges
Mangoes

Total: 50

Bar Chart

Pie Chart

Pie Chart Calculation: Each sector angle = \( \frac{f}{\sum f} \times 360° \)

Types of Statistical Diagrams

Bar Chart

Used for: Categorical/discrete data

Features:

  • Bars of equal width
  • Gaps between bars
  • Height represents frequency

Example: Favourite colours, types of transport

Pie Chart

Used for: Showing proportions of a whole

Features:

  • Circle divided into sectors
  • Angle proportional to frequency
  • Total = 360°

Formula: \( \text{Angle} = \frac{f}{\sum f} \times 360° \)

Histogram

Used for: Continuous/grouped data

Features:

  • Bars of equal width (for equal class intervals)
  • No gaps between bars
  • Area represents frequency

Example: Heights, weights, test scores in ranges

Frequency Polygon

Used for: Showing trends in grouped data

Features:

  • Points plotted at midpoint of each class
  • Points connected with straight lines
  • Can compare multiple distributions

Worked Example: Pie Chart Calculation

Calculating Pie Chart Angles

The table shows how 60 students travel to school. Calculate the angle for each sector.

Transport Frequency Calculation Angle
Walk 15 \( \frac{15}{60} \times 360° \) 90°
Bus 25 \( \frac{25}{60} \times 360° \) 150°
Car 12 \( \frac{12}{60} \times 360° \) 72°
Bicycle 8 \( \frac{8}{60} \times 360° \) 48°
Total 60 360°

Check: The angles should always add up to 360°. Here: \(90° + 150° + 72° + 48° = 360°\) ✓

Reading from Diagrams

CSEC-Style Question: Reading a Histogram

The histogram below shows the marks of 40 students on a test.

Questions:

  1. How many students scored between 40 and 50?
  2. What is the modal class?
  3. How many students scored 60 or more?

Answers:

  1. Reading from the histogram: 8 students scored between 40-50
  2. The modal class is the class with the highest frequency: 50-60 (12 students)
  3. Students scoring 60 or more: 60-70 (7) + 70-80 (3) = 10 students

Practice Questions

Test Your Understanding

1
In a pie chart, if a category has frequency 20 out of a total of 80, what angle should its sector have?
20°
80°
90°
100°
Solution: \( \text{Angle} = \frac{20}{80} \times 360° = \frac{1}{4} \times 360° = 90° \)
2
Which diagram is most appropriate for showing how marks are distributed in ranges (e.g., 0-10, 11-20, etc.)?
Pie chart
Bar chart
Histogram
Line graph
Explanation: Histograms are used for continuous data that has been grouped into class intervals. The key difference from bar charts is that histograms have no gaps between bars because the data is continuous.
3
A pie chart sector has an angle of 45°. If the total frequency is 200, what is the frequency for this sector?
45
25
8
50
Solution: Rearranging the formula: \( f = \frac{\text{Angle}}{360°} \times \text{Total} = \frac{45}{360} \times 200 = \frac{1}{8} \times 200 = 25 \)

Key Points to Remember

  • Frequency = how many times a value occurs
  • Bar charts have gaps (categorical data); histograms don't have gaps (continuous data)
  • Pie chart angle formula: \( \frac{f}{\sum f} \times 360° \)
  • Always check: pie chart angles must sum to 360°
  • The modal class is the class with the highest frequency
CSEC Examination Tips
  • Drawing pie charts: Use a protractor carefully and label each sector
  • Reading histograms: Remember that frequency = height × width (but for equal class widths, just read the height)
  • Show your working: Write down the formula and substitution for pie chart calculations
  • Check totals: Frequencies should add up to the given total, angles should sum to 360°
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