Frequency density Revision Notes for Edexcel GCSE Statistics

Frequency density

What are histograms with unequal class widths?

When dealing with data that has unequal class intervals (different sized groups), we cannot simply use frequency on the vertical axis like we do with regular histograms. Instead, we must use frequency density on the vertical axis.

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The key principle to understand is that in any histogram, the area of each bar represents the number of items (such as students, people, etc.) in that class. When class widths are different, using frequency alone would make the bars misleading because wider classes would appear more prominent than they should be.

The golden rule for frequency density

chatImportant

This is the most important formula you need to memorise:

Frequency density = frequency ÷ class width

$\text{Frequency density} = \frac{\text{frequency}}{\text{class width}}$

This formula ensures that when you multiply the height (frequency density) by the width (class width), you get the frequency - which means the area still represents the actual number in each class.

Understanding class width calculation

Before you can calculate frequency density, you need to work out the class width for each interval. This is simply:

$\text{Class width} = \text{upper boundary} - \text{lower boundary}$

For example:

For the interval $0 \leq t < 5$ : class width = $5 - 0 = 5$
For the interval $5 \leq t < 10$ : class width = $10 - 5 = 5$
For the interval $10 \leq t < 20$ : class width = $20 - 10 = 10$
For the interval $20 \leq t < 40$ : class width = $40 - 20 = 20$

Worked example: train lateness data

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Worked Example: Train Lateness Analysis

Let's work through a complete example step by step. We have data about how late some trains were:

Time, t (min)	Frequency
0 ≤ t < 5	12
5 ≤ t < 10	18
10 ≤ t < 20	14
20 ≤ t < 40	12

Step 1: Calculate each class width

$0 \leq t < 5$ : class width = $5 - 0 = 5$
$5 \leq t < 10$ : class width = $10 - 5 = 5$
$10 \leq t < 20$ : class width = $20 - 10 = 10$
$20 \leq t < 40$ : class width = $40 - 20 = 20$

Step 2: Use the golden rule to calculate frequency density

$0 \leq t < 5$ : frequency density = $12 \div 5 = 2.4$
$5 \leq t < 10$ : frequency density = $18 \div 5 = 3.6$
$10 \leq t < 20$ : frequency density = $14 \div 10 = 1.4$
$20 \leq t < 40$ : frequency density = $12 \div 20 = 0.6$

Step 3: Create your complete table

Time, t (min)	Frequency	Class width	Frequency density
0 ≤ t < 5	12	5	2.4
5 ≤ t < 10	18	5	3.6
10 ≤ t < 20	14	10	1.4
20 ≤ t < 40	12	20	0.6

Drawing the histogram

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When drawing your histogram, follow these essential steps:

Work out each class width (as shown above)
Use the golden rule to calculate frequency density for each class
Label the vertical axis as "Frequency density"
Decide on appropriate scales for both axes (these may be given in the exam)
Draw the bars with heights equal to the frequency density values

Notice how the bars get lower as the class widths get wider, even though some classes might have similar frequencies. This is because the frequency density adjusts for the different widths.

Common exam traps to avoid

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Critical Mistakes to Avoid:

Don't use frequency on the vertical axis - it must be frequency density for unequal class widths
Always check your class width calculations - be careful with inequalities like $\leq$ and $<$
Remember the formula - frequency density = frequency ÷ class width (not the other way around!)
Show your working - examiners want to see each step of your calculation
Check your units - if time is in minutes, make sure your class widths are also in minutes

Why does this method work?

The reason we use frequency density is mathematical: when you calculate the area of each bar (height × width), you get:

$\text{Area} = \text{frequency density} \times \text{class width} = \frac{\text{frequency}}{\text{class width}} \times \text{class width} = \text{frequency}$

This means the area of each bar still correctly represents the actual frequency, just as it should in any histogram.

bookmarkSummary

Key Points to Remember:

Golden rule: $\text{Frequency density} = \frac{\text{frequency}}{\text{class width}}$
Class width = upper boundary - lower boundary
The area of each bar represents the frequency (number of items)
Always label the vertical axis as "Frequency density"
Show all working in your calculations for full marks

Frequency density (Edexcel GCSE Statistics): Revision Notes