Lesson 4: Cumulative Frequency and Ogive

👩🏽‍🏫 Introduction

Imagine a teacher wants to know how many students scored below 30 marks on a test. Looking at individual frequencies might not help, but cumulative frequency and an ogive (a special type of graph) will! 📈

In this lesson, we’ll break down how to build cumulative frequency tables, draw ogives, and use them to estimate medians and percentiles — essential tools for exams, data analysis, and everyday insights!

• 🧮 Build cumulative frequency tables
• 📈 Draw an ogive (cumulative frequency graph)
• 📐 Use the ogive to estimate median and quartiles

🧠 Key Concepts

📌 1. Cumulative Frequency (CF)

• Cumulative frequency adds up the frequencies from top to bottom in a class interval table.
• It tells us how many values fall below a certain boundary.

📌 2. Cumulative Frequency Table

Let’s start with this data set:

Now add the cumulative frequency column:

📌 3. Ogive (Cumulative Frequency Curve)

• An ogive is a graph showing how data accumulates across intervals.
• You plot upper class boundaries on the x-axis and cumulative frequencies on the y-axis.
• Connect the points to form a smooth curve or line.

Ogive Graph for This Example:

🔍 Worked Example – Estimating the Median

1. 1. Step 1: Total frequency = 30 →
      
      Find the class where the 15th value falls → That’s the class 20–29

1. Step 2: Use linear interpolation or read from graph
   Median ≈ 29.5 or slightly below, depending on graph curve

✍🏽 Practice Exercises

🅰️ Section A – Complete the Table

1. Fill in the cumulative frequency column:
   

   Class Interval	Frequency	Cumulative Frequency
   0–10	4	?
   10–20	6	?
   20–30	10	?
   30–40	5	?

🅱️ Section B – Ogive Plotting

1. Plot an ogive using upper class boundaries (10, 20, 30, 40) and your completed cumulative frequency values.

Practice Ogive Graph:

🅾️ Section C – Interpretation

1. Estimate the median from your ogive graph.

✅ Answers and Explanations

• A1: CF values = 4, 10, 20, 25
• B1: Plot points: (10, 4), (20, 10), (30, 20), (40, 25)
• C1: Median = middle of 25 =
  → Look for where CF = 12.5 on y-axis → Median ≈ 22–24

📌 Recap

• 🧮 Cumulative frequency adds up data progressively across intervals
• 📈 An ogive graph helps visualize the total build-up of data
• 📐 Ogives are great for estimating medians and quartiles

💭 Reflection Prompt

If you recorded your study time in weekly ranges (e.g. 0–1 hr, 2–3 hrs…), how could you use an ogive to see how many days you studied more than 3 hours?