Cumulative Frequency Quizzes

Introduction

Cumulative frequency is an important concept in GCSE Maths statistics. It allows students to understand how data accumulates across intervals and helps in analysing distributions, finding medians, quartiles, and percentiles. Mastering cumulative frequency is essential for solving exam questions and interpreting real-world data.

Core Concepts

What is Cumulative Frequency?

Cumulative frequency is the running total of frequencies up to a certain class or value. It tells us how many data points fall below the upper boundary of a class. This is particularly useful for continuous data and for constructing cumulative frequency graphs (ogives).

Key Terms

• Frequency: Number of data points in a class interval.
• Cumulative Frequency (CF): Total number of data points up to a certain class. Calculated by adding the frequency of the current class to the cumulative frequency of the previous class.
• Class Interval: A range of values grouped together. Example: 0–5, 5–10, 10–15.
• Upper Class Boundary: The highest value in a class interval (used for plotting CF graphs).

Why Use Cumulative Frequency?

Cumulative frequency helps us to:

• Determine the median, quartiles, and percentiles.
• Understand the spread and distribution of data.
• Visualise data trends using cumulative frequency graphs (ogives).

Rules & Steps for Calculating Cumulative Frequency

1. List the class intervals and frequencies in a table.
2. Start with the first class: its cumulative frequency equals its frequency.
3. For each subsequent class, add its frequency to the cumulative frequency of the previous class:
   CF_current = CF_previous + Frequency_current
4. Continue until all classes are included. The final CF equals the total number of data points.

Worked Examples

Example 1: Basic Cumulative Frequency

Data on number of hours studied by students in a week:

Hours	Frequency
0–2	3
3–5	7
6–8	10
9–11	5

Step 1: Start with first class:

• CF for 0–2 = 3

Step 2: Add next frequencies cumulatively:

• CF for 3–5 = 3 + 7 = 10
• CF for 6–8 = 10 + 10 = 20
• CF for 9–11 = 20 + 5 = 25

The cumulative frequency table becomes:

Hours	Frequency	Cumulative Frequency
0–2	3	3
3–5	7	10
6–8	10	20
9–11	5	25

Example 2: Plotting a Cumulative Frequency Graph (Ogive)

To create a cumulative frequency graph:

1. Use the upper class boundaries on the x-axis (e.g., 2, 5, 8, 11).
2. Use cumulative frequency on the y-axis.
3. Plot points (upper boundary, CF) for each class:
• (2, 3)
• (5, 10)
• (8, 20)
• (11, 25)
4. Join the points with a smooth curve to complete the ogive.

Example 3: Finding Median from Cumulative Frequency

Use the previous CF table:

• Total number of students = 25
• Median position = \( \frac{25}{2} = 12.5^{\text{th}} \) value

Locate the class interval containing the 12.5th value. From CF table, 6–8 hours class contains the 12.5th student (CF = 10 for previous class, CF = 20 for current class).

Use the median formula:

$$ \text{Median} = L + \frac{\frac{N}{2} - CF_{\text{previous}}}{f} \times w $$

Where:

• \( L \) = lower boundary of median class = 6
• \( N \) = total frequency = 25
• \( CF_{\text{previous}} \) = 10
• \( f \) = frequency of median class = 10
• \( w \) = class width = 3

$$ \text{Median} = 6 + \frac{12.5 - 10}{10} \times 3 = 6 + 0.25 \times 3 = 6 + 0.75 = 6.75 $$

The median study time is 6.75 hours.

Example 4: Finding Quartiles

Quartiles divide data into four equal parts:

• Q1 = \( \frac{N}{4}^{\text{th}} \) value = 6.25th → in 3–5 class
• Q2 = Median = 6.75 (calculated above)
• Q3 = \( \frac{3N}{4}^{\text{th}} \) value = 18.75th → in 6–8 class

Use the same formula for Q1 and Q3 to find precise values.

Common Mistakes

• Using class midpoints instead of cumulative frequencies for plotting.
• Incorrectly identifying the median or quartile class interval.
• Forgetting to calculate the cumulative frequency for the first class.
• Plotting points at the wrong boundaries (always use upper class boundary).

Applications

Cumulative frequency is widely used in exams and real-life situations:

• Exam analysis – finding median, Q1, Q3 scores.
• Business – cumulative sales or customer visits over time.
• Science – cumulative rainfall, temperature data, or experiment results.

Strategies & Tips

• Always prepare a CF table before plotting graphs or calculating median/quartiles.
• Check total frequency to ensure CF is accurate.
• Use smooth curves for ogives rather than jagged lines.
• Label axes clearly and mark upper class boundaries correctly.
• Practice interpreting median, quartiles, and percentiles directly from cumulative frequency graphs.

Summary & Encouragement

Cumulative frequency is a powerful tool for analysing data distributions. Remember:

• CF is a running total of frequencies.
• Upper class boundaries are essential for plotting graphs.
• Use CF tables to calculate median, quartiles, and percentiles accurately.
• Always check total frequencies and boundaries for correctness.

Practice drawing cumulative frequency tables and graphs, calculating medians and quartiles, and interpreting data trends. This will solidify your understanding and improve exam performance. Try the quizzes now to reinforce these skills!