What Makes Histograms Different from Bar Charts?
Many students confuse histograms with bar charts, but they are fundamentally different. In a bar chart, the height of each bar represents the frequency. In a histogram, the area of each bar represents the frequency. This distinction matters whenever the class intervals (groups) have different widths.
Histograms with unequal class widths are a staple of the IGCSE Extended syllabus, appearing on Paper 2 and Paper 4. They typically carry between four and six marks and require you to either draw a histogram from given data or read information from a given histogram.
Understanding Frequency Density
Frequency density is the key concept that links frequency to area in a histogram:
Frequency density = Frequency ÷ Class width
Or equivalently:
Frequency = Frequency density × Class width
Since the area of each bar equals its width times its height, and we want area to represent frequency, the height must represent frequency density.
Why Not Just Use Frequency?
When class widths are equal, frequency and frequency density give bars with the same relative heights, so it makes no difference. But when class widths vary — which is the case in almost every IGCSE histogram question — using frequency for the height would be misleading. A wider class would appear more dominant simply because it covers a larger range, not necessarily because it has more data points.
Frequency density corrects for this by adjusting the height to account for the width.
Drawing a Histogram Step by Step
Step 1: Create a Frequency Density Table
Start by adding columns for class width and frequency density to the given data.
Example data:
| Time (t minutes) | Frequency |
|---|---|
| 0 < t ≤ 10 | 8 |
| 10 < t ≤ 20 | 15 |
| 20 < t ≤ 30 | 20 |
| 30 < t ≤ 50 | 24 |
| 50 < t ≤ 80 | 9 |
Add class width and frequency density:
| Time (t minutes) | Frequency | Class Width | Frequency Density |
|---|---|---|---|
| 0 < t ≤ 10 | 8 | 10 | 0.8 |
| 10 < t ≤ 20 | 15 | 10 | 1.5 |
| 20 < t ≤ 30 | 20 | 10 | 2.0 |
| 30 < t ≤ 50 | 24 | 20 | 1.2 |
| 50 < t ≤ 80 | 9 | 30 | 0.3 |
Step 2: Draw the Axes
- The horizontal axis shows the continuous variable (time, in this example) with a proper scale
- The vertical axis is labelled frequency density (never just “frequency” when class widths are unequal)
Step 3: Draw the Bars
- Each bar starts and ends at the class boundaries
- The height of each bar equals the frequency density
- Bars must touch (no gaps) because the data is continuous
Common mistakes at this stage:
- Leaving gaps between bars (this is not a bar chart)
- Labelling the vertical axis as “frequency” instead of “frequency density”
- Using the midpoint of each class on the horizontal axis instead of the actual boundaries
Reading a Histogram
When given a histogram and asked to find frequencies, use the reverse formula:
Frequency = Frequency density × Class width
This means you are calculating the area of each bar.
Worked Example
A histogram shows bars with the following properties:
- Bar from 0 to 5: height 2.4
- Bar from 5 to 15: height 3.6
- Bar from 15 to 25: height 1.8
- Bar from 25 to 40: height 1.2
Find the frequency for each class:
- 0 to 5: frequency = 2.4 × 5 = 12
- 5 to 15: frequency = 3.6 × 10 = 36
- 15 to 25: frequency = 1.8 × 10 = 18
- 25 to 40: frequency = 1.2 × 15 = 18
Total frequency = 12 + 36 + 18 + 18 = 84
Estimating the Median from a Histogram
Some IGCSE questions ask you to estimate the median from a histogram. The method involves finding the class that contains the median and then interpolating within that class.
Step-by-Step Method
- Find the total frequency (sum all the bar areas)
- The median is the (n/2)th value
- Add up frequencies class by class until you pass the median position
- Use interpolation within the median class
Using the data above:
- Total frequency = 84, so the median is the 42nd value
- Class 0–5 contains values 1 to 12
- Class 5–15 contains values 13 to 48
- The 42nd value falls in the 5–15 class
To interpolate: we need the 42nd value, and the class 5–15 starts at the 13th value and ends at the 48th. The 42nd value is the (42 − 12) = 30th value into this class, out of 36 values. The class width is 10, so:
Estimated median = 5 + (30/36) × 10 = 5 + 8.33 = 13.3 (to 3 s.f.)
Estimating the Mean from a Histogram
To estimate the mean, you need the midpoint of each class and the frequency:
Estimated mean = Σ(midpoint × frequency) / Σ frequency
Using our original data:
| Class | Midpoint | Frequency | Midpoint × Frequency |
|---|---|---|---|
| 0–10 | 5 | 8 | 40 |
| 10–20 | 15 | 15 | 225 |
| 20–30 | 25 | 20 | 500 |
| 30–50 | 40 | 24 | 960 |
| 50–80 | 65 | 9 | 585 |
Estimated mean = (40 + 225 + 500 + 960 + 585) / (8 + 15 + 20 + 24 + 9) = 2310 / 76 = 30.4 (to 3 s.f.)
Finding Missing Frequencies
Some questions give you a partially completed histogram and a piece of additional information (such as the total frequency or the frequency for one class) and ask you to complete the histogram or find missing frequencies.
Strategy:
- Use the given information to find the scale on the frequency density axis
- Read the height of each bar
- Calculate frequencies using frequency = frequency density × class width
- Use any given totals to find missing values
Common Exam Pitfalls
- Using frequency instead of frequency density for bar heights. Always calculate frequency density when class widths are unequal.
- Getting the class width wrong. For example, the class 10 < t ≤ 25 has width 15, not 10.
- Misreading the scale. Check the vertical axis carefully — each small square might not represent 1 unit.
- Forgetting that area equals frequency. When reading values from a histogram, always multiply height by width.
- Not labelling axes correctly. The vertical axis must say “frequency density,” and the horizontal axis must show the actual variable with proper units.
Practice Plan
- Start by practising the conversion between frequency and frequency density tables
- Draw several histograms from data tables with unequal class widths
- Practise reading histograms to extract frequencies
- Attempt past paper questions combining histograms with median and mean estimation
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