What is a histogram?
A histogram looks similar to a bar chart, but it's used for continuous grouped data with unequal class widths. Because the classes have different widths, the height of each bar cannot simply be the frequency — instead, the height represents frequency density.
- Frequency density = frequency ÷ class width
- The height of each bar is the frequency density
- Crucially, the area of each bar (height × width) represents the frequency — not the height alone
- There are no gaps between the bars, since the data is continuous
Calculating frequency density
Worked Example 1
The table shows the weight of 81 parcels delivered by a courier. Calculate the frequency density for each class.
| Weight (kg) | 0–2 | 2–5 | 5–10 | 10–20 | 20–40 |
|---|---|---|---|---|---|
| Frequency | 8 | 18 | 25 | 20 | 10 |
| Class width | 2 | 3 | 5 | 10 | 20 |
| Frequency density | 4 | 6 | 5 | 2 | 0.5 |
1
0–2 kg: frequency density = 8 ÷ 2 = 4
2
2–5 kg: 18 ÷ 3 = 6. 5–10 kg: 25 ÷ 5 = 5. 10–20 kg: 20 ÷ 10 = 2. 20–40 kg: 10 ÷ 20 = 0.5
Answer4, 6, 5, 2, 0.5
Bar height = frequency density; bar area = frequency. The wide, short bar for 20–40 kg still represents 10 parcels, the same principle as the tall, narrow bar for 2–5 kg representing 18.
Reading a frequency from a histogram
Worked Example 2
Using the histogram above, find the number of parcels weighing between 5 kg and 10 kg.
1
Read the frequency density (height) of that bar: 5
2
Multiply by the class width: frequency = height × width = 5 × (10 − 5) = 5 × 5
3
= 25 parcels
Answer25 parcels
Finding the height of a new bar
Worked Example 3
A new class, 40–50 kg, needs to be added to the histogram, representing 6 parcels. Find the height (frequency density) of this bar.
1
Class width = 50 − 40 = 10
2
Frequency density = frequency ÷ width = 6 ÷ 10
3
= 0.6
Answer0.6
Estimating a frequency within part of a class
Worked Example 4 — Proportional area
Using the original data (10–20 kg class has frequency 20), estimate the number of parcels weighing between 10 kg and 15 kg.
1
Assume the 20 parcels are spread evenly across the 10–20 kg class (width 10)
2
10–15 kg is half of this class's width (5 out of 10)
3
Estimated frequency = 20 × (5 ÷ 10) = 10 parcels
Answer≈ 10 parcels
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A class interval has frequency 24 and width 6. Find the frequency density.Foundation
Q2A histogram bar has frequency density 3 and width 5. Find the frequency it represents.Foundation
Q3True or false: in a histogram, the height of each bar always represents the frequency. Explain your answer.Foundation
Q4A class interval 10–15 has frequency 20. Find its width and frequency density.Foundation
Q5A histogram bar has height (frequency density) 2.5 and represents 25 parcels. Find the width of the class.Foundation
Higher (Grade 5–7)
Q6
The table shows the time taken by 65 customers to be served. Find the frequency density for the 0–5 minute class.
Higher| Time (min) | 0–5 | 5–15 | 15–20 |
|---|---|---|---|
| Frequency | 15 | 40 | 10 |
Q7Using the data from Q6, find the frequency density for the 5–15 minute class.Higher
Q8Using the data from Q6, find the frequency density for the 15–20 minute class.Higher
Q9A histogram bar spans 20–50 (width 30) with height (frequency density) 1.2. Find the frequency for this class.Higher
Q10A histogram has a bar for 0–10 with frequency density 3, and another bar for 10–30 representing 24 people. Find the frequency density of the second bar.Higher
Higher — Hard (Grade 8–9)
Q11A class interval 20–40 has frequency 30. Estimate the number of values between 20 and 25, assuming they are evenly spread within the class.Grade 8–9
Q12A histogram bar for the class 10–16 has height 4.5. Find the frequency for this class.Grade 8–9
Q13A histogram bar has an area (frequency) of 45 and spans a class width of 9. Find the frequency density.Grade 8–9
Q14In a histogram, two adjacent bars have equal areas but different widths. What must be true about their heights?Grade 8–9
Q15Explain why frequency density, rather than frequency, must be used for the height of histogram bars when class widths are unequal.Grade 8–9
Answers
Foundation (Q1–Q5)
Q14(24 ÷ 6)
Q215(3 × 5)
Q3False(the height represents frequency density; the area represents frequency)
Q4Width = 5, frequency density = 4(20 ÷ 5)
Q510(25 ÷ 2.5)
Higher (Q6–Q10)
Q63(15 ÷ 5)
Q74(40 ÷ 10)
Q82(10 ÷ 5)
Q936(1.2 × 30)
Q101.2(24 ÷ 20; the first bar's height of 3 isn't needed for this calculation)
Higher — Hard (Q11–Q15)
Q11≈ 7.5(frequency density = 30 ÷ 20 = 1.5; 1.5 × 5)
Q1227(width = 6; 4.5 × 6)
Q135(45 ÷ 9)
Q14Their heights must be different, in inverse proportion to their widths — since area (frequency) is the same but width differs, the narrower bar must be taller and the wider bar shorter
Q15If raw frequency were used as the height, a wide class would automatically produce a tall bar even if its data wasn't especially dense — making it look more significant than it really is. Using frequency density (frequency ÷ width) instead means the bar's area represents frequency, keeping visual comparisons between classes of different widths fair and accurate
Common mistakes
Common Mistake 1
Using frequency as the height instead of frequency density
This is the single most common histogram error. Always calculate frequency ÷ class width before plotting the bar height — never plot the raw frequency directly when class widths are unequal.
Common Mistake 2
Reading the height off the graph as the frequency
When reading a histogram, the height gives the frequency density. You must multiply by the class width to get the actual frequency — height alone is not the answer.
Common Mistake 3
Using the wrong class width
Always calculate the class width as (upper boundary − lower boundary) for that specific class. Widths change between classes in a histogram with unequal intervals — don't assume they're all the same.
Common Mistake 4
Forgetting that estimates within a class assume even spread
Estimating a frequency for part of a class (e.g. "how many between 10 and 15" from a 10–20 class) relies on the assumption that the data is spread evenly across that class — always state this assumption when giving this kind of estimate.
Exam tips
💡 Exam Tip 1
Write the formula down first
Before any calculation, write "frequency density = frequency ÷ class width" (or the rearranged version you need) — this earns a method mark even if the final number is wrong, and keeps you from mixing up which value goes where.
💡 Exam Tip 2
Build a table before plotting
Set out class, frequency, class width and frequency density in a table first, exactly as shown in the worked examples — this makes errors much easier to spot than calculating each bar separately as you go.
💡 Exam Tip 3
Remember: area = frequency, always
Whether you're reading a bar or drawing one, "area represents frequency" is the one rule that never changes in a histogram — if a question feels confusing, go back to this rule and it usually becomes clear what to calculate.
💡 Exam Tip 4
Check your y-axis label says "frequency density"
When drawing a histogram, the vertical axis must be labelled "frequency density", not "frequency" — examiners specifically check for this, since it signals you understand what the bar heights represent.
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