Histograms and Frequency Density (Edexcel GCSE Maths): Revision Notes

Histograms and frequency density

What are histograms?

A histogram is a special type of bar chart where the bars can have different widths. This makes them different from regular bar charts where all bars have the same width. The varying widths might seem confusing at first, but they're actually very useful for showing grouped data with unequal class intervals.

The key thing to remember is that histograms don't show frequency directly on the vertical axis like regular bar charts do. Instead, they show something called frequency density.

Understanding frequency density

Frequency density is what we measure on the vertical axis of a histogram. It's not the same as frequency (which tells us "how many" items are in each group). Instead, frequency density takes into account both the frequency and the width of each class interval.

Think of it this way: if you have a narrow class interval with lots of data points, the frequency density will be high. If you have a wide class interval with the same number of data points, the frequency density will be lower.

Working with the frequency density formula

You can rearrange the frequency density formula to find different values depending on what information you have:

$\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$

This rearranged formula is particularly useful when you're reading information from a histogram and need to calculate the actual frequency for each class interval.

Step-by-step approach to histogram problems

When working with histograms, follow these systematic steps:

1. Add a frequency density column to your data table - Use the formula $\text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}$ to calculate the frequency density for each class interval.
2. Use the frequency densities to create or label your histogram - The frequency density values become the heights of the bars on your histogram.
3. Find missing information - If you have a histogram and need to find missing frequencies, use the rearranged formula: $\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$.

Creating histograms from data tables

When you're given a frequency table and asked to draw a histogram, you need to:

1. Calculate the class width for each interval - This is the difference between the upper and lower boundaries of each class.
2. Calculate the frequency density for each class - Apply the formula $\frac{\text{Frequency}}{\text{Class Width}}$ to each row.
3. Draw bars with heights equal to the frequency density values - The vertical position of each bar top corresponds to its frequency density.
4. Ensure the bars are drawn with the correct widths - The horizontal width of each bar must match the class interval width.

Length (mm)	Frequency
0 < x ≤ 40	20
40 < x ≤ 60	45
60 < x ≤ 65	15
65 < x ≤ 100	70

Reading and interpreting histograms

When interpreting a histogram, remember that:

• The vertical axis shows frequency density, not frequency
• The area of each bar represents the frequency for that class interval
• Taller bars don't necessarily mean higher frequencies if the class widths are different
• You need to multiply frequency density by class width to get the actual frequency

This is why understanding the relationship between frequency, frequency density, and class width is so important for correctly interpreting histogram data.