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Introduction

Cumulative frequency curves are an essential tool for GCSE Mathematics students. They help visualize and analyze data, making them a valuable study aid for probability, statistics, and data handling topics.

Key Concepts and Definitions

• Cumulative frequency: The total number of data points that fall at or below a given value.
• Cumulative frequency curve: A graph showing the relationship between the cumulative frequency and the corresponding values of the data.
• Median: The middle value of the data when arranged in ascending order.
• Interquartile range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1).

Constructing a Cumulative Frequency Curve

1. Arrange the data in ascending order.

2. Calculate the cumulative frequency for each data point: Cumulative Frequency = Frequency at Data Point + Cumulative Frequency of Previous Data Point.

3. Plot the pairs of values (data point, cumulative frequency) on a graph.

4. Connect the points with a smooth curve.

Interpreting a Cumulative Frequency Curve

• Finding the median: The value where the cumulative frequency curve reaches 50% (0.5) is the median.
• Finding quartiles: The value where the cumulative frequency curve reaches 25% (0.25) is the lower quartile (Q1). The value where the cumulative frequency curve reaches 75% (0.75) is the upper quartile (Q3).
• Estimating the range: The range can be estimated as the highest data point minus the lowest data point, or by finding the difference between the upper and lower quartiles.

Common Mistakes to Avoid

• Not starting the cumulative frequency at 0.
• Not connecting the points with a curve.
• Incorrectly interpolating the median or quartiles.

Worked Examples

• Example 1: Finding the Median

Data: {5, 7, 8, 9, 11}

Cumulative Frequency: {1, 2, 3, 4, 5}

Cumulative frequency curve:


The median is the value where the cumulative frequency curve reaches 0.5, which is 8.

• Example 2: Finding the IQR

Data: {3, 4, 6, 7, 8, 9, 10}

Cumulative Frequency: {1, 2, 3, 4, 5, 6, 7}

Cumulative frequency curve:


Q1 is the value where the cumulative frequency curve reaches 0.25, which is 6.

Q3 is the value where the cumulative frequency curve reaches 0.75, which is 9.

Therefore, the IQR is 9 - 6 = 3.

Practice Problems

• Problem 1:

Find the median and IQR of the following data: {4, 5, 6, 7, 9, 10, 12}

• Problem 2:

Construct a cumulative frequency curve for the following data: {20, 25, 30, 35, 40, 45, 50}

Exam Tips

• Practice drawing cumulative frequency curves for different data sets.
• Know how to use the curve to find the median, quartiles, and range.
• Check your work for errors, such as starting the cumulative frequency at 0 and connecting the points with a smooth curve.

FAQ

• Q: What is the difference between a cumulative frequency curve and a frequency histogram?

A: A cumulative frequency curve shows the cumulative frequency at each data point, while a frequency histogram shows the frequency of each data value.

• Q: How do I find the mean using a cumulative frequency curve?

A: The mean cannot be found directly from the cumulative frequency curve. You need the original data to calculate the mean.

• Q: What is the shape of a cumulative frequency curve for a normally distributed data set?

A: The cumulative frequency curve for a normally distributed data set will be symmetric and bell-shaped.