Histograms

Revise Histograms for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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📋Chapter Overview

1. Frequency Density

A measure of how tightly packed data is in a class interval. Calculated by dividing frequency by class width. Essential when class widths are unequal.

2. Drawing Histograms

Calculate frequency density for each class, then draw bars where height = FD and width = class width. Area of bar = frequency.

3. Interpreting Histograms

To find frequency from a histogram, multiply the height (FD) by the width (class width). Compare distributions using same scales and intervals.

📊1. Frequency Density

What Is Frequency Density?

Frequency density is a special measure designed to handle grouped continuous data where the class widths are unequal. Think of it like this: if you have 20 students in a class interval that spans 10 units, the data is "denser" than having 20 students in an interval that spans 50 units. Frequency density captures this idea by always accounting for interval width.

In a regular bar chart, we look at the height of the bar to read the frequency. But in a histogram, the height is frequency density, not frequency. The area of the bar tells you the frequency. This is a crucial distinction that many students miss.

Frequency Density Formula:

Frequency Density = Frequency ÷ Class Width

Or rearranged: Frequency = Frequency Density × Class Width

💡 Why we need it: When class intervals have different widths, you can't compare frequencies by bar height alone. A class with 50 items in a 5-unit interval is much denser than a class with 50 items in a 20-unit interval. Frequency density fixes this by normalizing to a consistent unit width.

Calculating Frequency Density

The method is straightforward, but you must be careful to show your working. Here's the step-by-step process:

Calculate the class width for each interval (upper boundary minus lower boundary)
Divide the frequency by the class width to get frequency density
Check: wider intervals should have smaller frequency densities (all else equal)

✏️ Worked Example: Train Speeds

The table shows the average speeds (in m/s) of trains travelling through a region. Calculate the frequency density for each class interval.

Speed (m/s)	Frequency	Class Width	Frequency Density
20 ≤ s < 40	5	40 – 20 = 20	5 ÷ 20 = 0.25
40 ≤ s < 50	15	50 – 40 = 10	15 ÷ 10 = 1.5
50 ≤ s < 55	28	55 – 50 = 5	28 ÷ 5 = 5.6
55 ≤ s < 60	38	60 – 55 = 5	38 ÷ 5 = 7.6
60 ≤ s < 70	14	70 – 60 = 10	14 ÷ 10 = 1.4

Key observation:

📝Practice Question 1

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Also in the full note

📐2. Drawing Histograms
🔍3. Interpreting Histograms
🧠What to Memorise
✅Concepts Checklist
⚡Exam Tips & Common Pitfalls
📌Typical Exam Question Patterns
Histogram vs Bar Chart — Key Differences
How to Draw a Histogram

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Histograms

Histograms

📋Chapter Overview

1. Frequency Density

2. Drawing Histograms

3. Interpreting Histograms

📊1. Frequency Density

What Is Frequency Density?

Calculating Frequency Density

✏️ Worked Example: Train Speeds

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