Statistics
Grade 7-9Histograms
A histogram is used for grouped data with unequal class widths, where the height of each bar is not the frequency itself but the frequency density, so that the area of each bar represents the frequency instead. This lesson covers calculating frequency density, finding a frequency from a bar's height, finding a missing frequency using the total, and estimating a frequency for part of a class.
What you need to know
- Histograms are for grouped continuous data and have no gaps between the bars.
- When class widths are unequal, the vertical axis is frequency density, not frequency.
- Frequency density is frequency divided by class width.
- The frequency for a class is the area of its bar, not its height.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Calculating frequency density
Frequency density is the frequency divided by the class width. This is needed because histogram classes are not always the same width, so a bar's height alone would not fairly represent how many items are in it.
24 people took between 20 and 30 minutes to complete a fun run, a class width of 10. The frequency density is 24 ÷ 10 = 2.4.
Finding a frequency from a bar
Rearranging the frequency density formula gives frequency = frequency density × class width. Read the height of the bar from the vertical axis, then multiply by the width of that class.
A bar has a frequency density of 1.0 over the 30 to 50 minute class, a width of 20. The frequency is 1.0 × 20 = 20 people.
Finding a missing frequency
If one bar is missing from the histogram but the total frequency is known, add up the frequencies of all the other classes using frequency density × class width, then subtract this total from the overall total.
With a total of 80 people and known frequencies of 8, 20, 24 and 20 for four classes, these sum to 72, so the missing class has a frequency of 80 − 72 = 8 people.
Estimating a frequency for part of a class
Assuming the data is spread evenly within a class, find the frequency density for the whole class first, then multiply by just the width of the part being asked about.
The 20 to 30 minute class has a frequency density of 2.4. To estimate the number of people who took between 20 and 25 minutes, a width of 5, multiply 2.4 × 5 = 12 people.
Worked examples
Three exam-style questions, fully solved
The histogram shows the time taken by 80 people to complete a fun run. 24 people took between 20 and 30 minutes. Calculate the frequency density for this class.
Easy- 1.Find the class width: 30 − 20 = 10
- 2.Divide the frequency by the class width: 24 ÷ 10
Answer: 2.4
The histogram shows the time taken by 80 people to complete a fun run, but the bar for the 50-70 minute class is missing. The other four classes have frequencies of 8, 20, 24 and 20. Find the number of people who took between 50 and 70 minutes.
Medium- 1.Add the known frequencies: 8 + 20 + 24 + 20 = 72
- 2.Subtract from the total: 80 − 72
Answer: 8 people
The histogram shows the time taken by 80 people to complete a fun run. The 20-30 minute class has a frequency density of 2.4. Assuming the data is spread evenly within each class, estimate the number of people who took between 20 and 25 minutes.
Hard- 1.Find the width of the part being asked about: 25 − 20 = 5
- 2.Multiply the frequency density by this width: 2.4 × 5
Answer: 12 people
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A histogram bar covers the class 10–20 and represents a frequency of 24. Work out its frequency density.
A histogram bar covers the class 0–5 and has a frequency density of 3. Work out its frequency.
A histogram bar covers the class 20–40 with a frequency density of 1.5. Work out its frequency.
A histogram bar represents a frequency of 18 and has a frequency density of 6. Work out the class width.
On a histogram, the class 5–15 has a frequency density of 4. Work out the frequency of this class.
A histogram's class 30–50 represents a frequency of 50. Work out the frequency density.
Avoid these
The mistakes students make most often
Confusing frequency density with frequency itself, forgetting that the height of a histogram bar is not the frequency when class widths are unequal.
Forgetting to multiply by the class width when converting a frequency density back to a frequency.
Using the wrong class width, especially when different classes in the same histogram have different widths.
When estimating a frequency for part of a class, using the whole class width instead of just the width of the part being asked about.
When finding a missing frequency, forgetting to convert every other bar's frequency density to an actual frequency before subtracting from the total.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
Teachably is a GCSE Maths tutoring practice
This hub is our free resource and it stays free. If you would like one-to-one help alongside it, a short call is the best place to start. There is also a free video course for parents.