Histograms [H]

This lesson covers histograms — a Higher tier topic on the AQA GCSE Mathematics specification. Unlike bar charts, histograms use frequency density on the vertical axis and can have bars of unequal width. Understanding how to draw, read, and interpret histograms is essential for the Higher paper.

---

How Histograms Differ from Bar Charts

Feature	Bar Chart	Histogram
Vertical axis	Frequency	Frequency density
Bar width	All the same	Can vary (based on class width)
Gaps between bars	Yes (for discrete/categorical data)	No gaps
What represents the frequency	Height of the bar	Area of the bar
Data type	Discrete or categorical	Continuous

The crucial difference is that in a histogram, the area of each bar represents the frequency, not the height.

---

Frequency Density

Frequency density = frequency / class width

Or equivalently:

Frequency = frequency density x class width

The class width is the difference between the upper and lower boundaries of the class interval.

Worked Example — Calculating Frequency Density

The table shows the ages of 50 people at a concert.

Age (a years)	Class Width	Frequency	Frequency Density
10 < a ≤ 20	10	8	8 / 10 = 0.8
20 < a ≤ 25	5	12	12 / 5 = 2.4
25 < a ≤ 30	5	15	15 / 5 = 3.0
30 < a ≤ 40	10	10	10 / 10 = 1.0
40 < a ≤ 60	20	5	5 / 20 = 0.25

Notice how the class widths are not all the same — this is why we must use frequency density rather than frequency on the vertical axis.

Exam Tip: Always calculate the class width first. For the class 20 < a ≤ 25, the class width is 25 - 20 = 5 (not 6 — do not count the endpoints, just subtract). A common mistake is getting the class width wrong, which leads to incorrect frequency densities.

---

Drawing a Histogram

To draw a histogram:

1. Calculate the class width for each interval.
2. Calculate the frequency density for each class (frequency / class width).
3. Draw bars with the correct width (matching the class interval on the horizontal axis).
4. The height of each bar equals the frequency density.
5. There should be no gaps between the bars.
6. Label the horizontal axis with the continuous variable and the vertical axis "Frequency density".

Key Points When Drawing

• The horizontal axis must have a continuous, linear scale (not categories).
• Bars must be drawn at the correct position on the scale, not evenly spaced.
• The width of each bar must match the class width exactly.

graph LR
    A[Grouped Frequency Table] --> B[Calculate Class Widths]
    B --> C[Calculate Frequency Density]
    C --> D[Draw Bars]
    D --> E[Width = Class Width]
    D --> F[Height = Frequency Density]
    D --> G[Area = Frequency]

Exam Tip: When drawing a histogram in the exam, use a ruler and be precise with widths and heights. The bars must start and end at the correct class boundaries. A bar from 20 to 25 must be exactly 5 units wide on the horizontal scale.

---

Reading a Histogram

When given a histogram, you may need to find frequencies. Remember:

Frequency = frequency density x class width

This is the area of the bar.

Worked Example

A histogram shows the weights of parcels. One bar covers the interval 2 < w ≤ 5 and has a height (frequency density) of 4.

Class width = 5 - 2 = 3

Frequency = 4 x 3 = 12 parcels

Worked Example — Finding All Frequencies

A histogram shows the following:

Weight (w kg)	Class Width	Frequency Density	Frequency
0 < w ≤ 2	2	3	3 x 2 = 6
2 < w ≤ 5	3	4	4 x 3 = 12
5 < w ≤ 10	5	2.4	2.4 x 5 = 12
10 < w ≤ 20	10	1.2	1.2 x 10 = 12
20 < w ≤ 25	5	0.8	0.8 x 5 = 4

Total frequency = 6 + 12 + 12 + 12 + 4 = 46 parcels

---

Finding a Missing Bar

Sometimes you are given a histogram with a missing bar and told the frequency for that class. You need to calculate the frequency density and draw the bar.

Worked Example

A histogram shows the times (t seconds) taken to run 100m. All bars are drawn except for the class 13 < t ≤ 14, which has a frequency of 9.

Class width = 14 - 13 = 1

Frequency density = 9 / 1 = 9

Draw a bar from 13 to 14 on the horizontal axis with a height of 9 on the frequency density axis.

Exam Tip: If a question gives you one frequency and asks you to complete a histogram, first use that frequency to work out the scale on the frequency density axis. Divide the frequency by the class width to get the frequency density, then check it matches the height of the corresponding bar. This lets you read the frequency densities of the other bars and calculate their frequencies.

---

Using a Histogram to Estimate Values

Estimating the Number of Values in a Given Range

Sometimes you need to estimate how many values fall within a range that does not match the class boundaries.

Worked Example

Using the weights table above, estimate the number of parcels weighing between 3 and 7 kg.

This range spans two classes:

From the class 2 < w ≤ 5 (frequency density = 4): The portion from 3 to 5 has width 2. Frequency = 4 x 2 = 8

From the class 5 < w ≤ 10 (frequency density = 2.4): The portion from 5 to 7 has width 2. Frequency = 2.4 x 2 = 4.8

Total estimate = 8 + 4.8 = 12.8 (approximately 13 parcels)

This is only an estimate because we assume the data is evenly spread within each class.

---

Building a Frequency Table from a Histogram

If given a histogram, you can reconstruct the frequency table:

1. Read the class boundaries from the horizontal axis.
2. Read the frequency density (height) for each bar.
3. Calculate the class width for each bar.
4. Calculate frequency = frequency density x class width.

Worked Example

A histogram has the following bars: