Why Histograms Are Different from Bar Charts
Many students confuse histograms with bar charts, and this confusion costs marks. The key difference is that in a histogram, the area of each bar represents the frequency, not the height. This matters because histograms are used for continuous data with unequal class widths. If you simply use the height to represent frequency when class widths differ, the visual representation is misleading.
To handle unequal class widths, we use frequency density on the y-axis instead of frequency. This is the single most important concept in histogram questions.
The Formula
Frequency density = Frequency ÷ Class width
Equivalently: Frequency = Frequency density × Class width
This second form is what you use when reading values from a histogram — you calculate the area of each bar.
The Problem
The table shows the times (in minutes) taken by 80 students to complete a test:
| Time (t minutes) | Frequency |
|---|---|
| 20 < t ≤ 30 | 8 |
| 30 < t ≤ 40 | 16 |
| 40 < t ≤ 50 | 24 |
| 50 < t ≤ 60 | 20 |
| 60 < t ≤ 80 | 10 |
| 80 < t ≤ 120 | 2 |
(a) Calculate the frequency density for each class. (b) Draw the histogram. (c) Estimate the number of students who took between 35 and 55 minutes.
Part (a): Calculating Frequency Density
For each class, divide the frequency by the class width:
| Time (t minutes) | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 20 < t ≤ 30 | 8 | 10 | 0.8 |
| 30 < t ≤ 40 | 16 | 10 | 1.6 |
| 40 < t ≤ 50 | 24 | 10 | 2.4 |
| 50 < t ≤ 60 | 20 | 10 | 2.0 |
| 60 < t ≤ 80 | 10 | 20 | 0.5 |
| 80 < t ≤ 120 | 2 | 40 | 0.05 |
Notice how the last two classes have different widths (20 and 40 minutes respectively) compared to the first four (10 minutes each). This is why we need frequency density — without it, the bars for these wider classes would be misleadingly tall.
Part (b): Drawing the Histogram
When drawing the histogram:
- The x-axis shows the continuous variable (time, in minutes) with a continuous scale
- The y-axis shows frequency density
- Each bar spans the full width of its class interval
- There are no gaps between bars (unlike bar charts)
The bars would be:
- From 20 to 30: height 0.8
- From 30 to 40: height 1.6
- From 40 to 50: height 2.4
- From 50 to 60: height 2.0
- From 60 to 80: height 0.5 (this bar is twice as wide)
- From 80 to 120: height 0.05 (this bar is four times as wide)
Check: the total area of all bars should equal the total frequency. Let us verify:
- 10 × 0.8 = 8
- 10 × 1.6 = 16
- 10 × 2.4 = 24
- 10 × 2.0 = 20
- 20 × 0.5 = 10
- 40 × 0.05 = 2
Total = 8 + 16 + 24 + 20 + 10 + 2 = 80. Correct — this matches the total number of students.
Part (c): Estimating Students Between 35 and 55 Minutes
This is where the area interpretation becomes essential. We need to estimate the frequency for the interval 35 < t ≤ 55, which spans parts of two complete classes and one full class.
From 35 to 40: This is half of the 30-40 class (5 out of 10 minutes). We assume the data is uniformly distributed within each class.
- Estimated frequency = 0.5 × 16 = 8
From 40 to 50: This is the entire 40-50 class.
- Frequency = 24
From 50 to 55: This is half of the 50-60 class (5 out of 10 minutes).
- Estimated frequency = 0.5 × 20 = 10
Total estimate: 8 + 24 + 10 = 42 students
Alternatively, using frequency density: calculate the area of the bars between 35 and 55.
- Area from 35 to 40 = 5 × 1.6 = 8
- Area from 40 to 50 = 10 × 2.4 = 24
- Area from 50 to 55 = 5 × 2.0 = 10
- Total area = 42
Both methods give the same answer, confirming that approximately 42 students took between 35 and 55 minutes.
Reading a Histogram When No Table Is Given
Sometimes the exam gives you a histogram and asks you to find frequencies or complete a table. The process is:
- Read the frequency density from the y-axis for each bar
- Measure the class width from the x-axis for each bar
- Calculate frequency = frequency density × class width for each bar
This is a common Part (a) that precedes further analysis.
Why the Uniform Distribution Assumption Matters
When we estimated the number of students between 35 and 55 minutes, we assumed that within each class, the students are evenly spread. This is an approximation — in reality, they might be clustered towards one end of the class. But for IGCSE purposes, this assumption is standard and the examiner expects you to use it.
The word “estimate” in the question is a deliberate signal that you should use this approach. If the question said “find” instead, it would mean you have exact data.
Common Mistakes
- Using frequency on the y-axis instead of frequency density. This is the number one error. Always check whether the class widths are equal. If they are not, you must use frequency density.
- Calculating class width incorrectly. The class 60 < t ≤ 80 has a width of 20, not 80 or 60. Class width = upper boundary − lower boundary.
- Leaving gaps between bars. Histograms represent continuous data, so bars must touch. Gaps suggest the data is discrete, which is incorrect for histograms.
- Not labelling axes correctly. The y-axis must be labelled “frequency density,” not “frequency.” The x-axis must show the continuous variable with a proper scale.
- Estimating frequencies without proportional reasoning. When asked for a frequency within part of a class, you must calculate the proportion of the class width that falls within the required range, then multiply by the class frequency.
Histogram vs. Bar Chart: A Quick Comparison
- Bar chart: Used for discrete or categorical data. Height represents frequency. Bars may have gaps.
- Histogram: Used for continuous grouped data. Area represents frequency. Bars have no gaps. Y-axis shows frequency density when class widths are unequal.
If all class widths are equal, the histogram looks identical to a bar chart (because frequency density is proportional to frequency). The distinction only becomes visible and important when class widths differ.
Practice Question
A histogram shows the masses of 100 parcels. The bar from 0 to 2 kg has a height (frequency density) of 5. The bar from 2 to 5 kg has a height of 12. The bar from 5 to 10 kg has a height of 6. The bar from 10 to 20 kg has a height of 1.
Find the frequency for each class and verify the total is 100. Then estimate how many parcels weigh between 3 and 8 kg.
Get Expert Help with IGCSE Maths
If histograms — or any other statistics topic — are giving you trouble, a specialist IGCSE Maths tutor can help you build confidence and technique in just a few sessions.
Need Help With IGCSE Maths?
Book a free 60-minute trial class with Teacher Rig and get personalised guidance for your IGCSE Maths preparation.