Histograms and Frequency Density
A histogram displays grouped continuous data where the class widths are unequal, and the crucial difference from a bar chart is that the FREQUENCY is represented by the AREA of each bar, not by its height. The vertical axis is frequency density, defined as frequency divided by class width. This is what makes a histogram honest when classes have different widths: without it, a wide class would look more numerous simply for being wide. Reading a histogram reverses the calculation, since frequency equals frequency density multiplied by class width, which is the area of the bar. Estimating the number of values in part of a class assumes the data is spread evenly across that class.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Work out the class width for every group first, subtracting the lower boundary from the upper boundary, and be careful with boundaries written as 10 less than x less than or equal to 20, where the width is 10.
- Calculate frequency density as frequency divided by class width, and label the vertical axis frequency density, not frequency.
- Draw each bar to the height of its frequency density, with no gaps between bars, because the data is continuous.
- To read a frequency from a histogram, multiply the frequency density by the class width, which is the same as finding the area of the bar.
- To estimate the number of values in part of a class, find what fraction of that class width the part covers and take the same fraction of its frequency, stating the assumption that the values are evenly spread across the class.
- Check the total: the areas of all the bars should sum to the total frequency, which is a quick way to catch an arithmetic slip.
Worked example
A histogram shows times in minutes. The class 10 less than t less than or equal to 30 has frequency density 1.5, and the class 30 less than t less than or equal to 40 has frequency density 4. Find the frequency of each class, and estimate how many values lie between 20 and 30 minutes.
- For the first class, the width is 30 - 10 = 20 minutes.
- Frequency = frequency density times class width = 1.5 times 20 = 30 values.
- For the second class, the width is 40 - 30 = 10 minutes, so the frequency = 4 times 10 = 40 values.
- For the estimate between 20 and 30, note this is part of the first class, which runs from 10 to 30.
- The part from 20 to 30 is 10 minutes out of a 20-minute class, that is one half of it.
- So the estimate is one half of 30 = 15 values, assuming the values are evenly spread across the class from 10 to 30.
Practice questions
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Q1What quantity is plotted on the vertical axis of a histogram?Show answer
Answer: Frequency density.
Q2Give the formula for frequency density.Show answer
Answer: Frequency divided by class width.
Q3A class 0 less than x less than or equal to 25 has frequency 50. Find its frequency density.Show answer
Answer: 50 divided by 25 = 2.
Q4A bar has frequency density 3.5 and class width 8. Find the frequency.Show answer
Answer: 3.5 times 8 = 28.
Q5What does the AREA of a bar in a histogram represent?Show answer
Answer: The frequency of that class.
Q6Why are histograms used instead of bar charts for unequal class widths?Show answer
Answer: Because plotting frequency as height would make a wide class look more numerous simply because it is wide. Using area removes that distortion.
Q7What assumption is made when estimating the frequency in part of a class?Show answer
Answer: That the values are distributed evenly across the whole class.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
The table shows the masses of 100 parcels. 0 less than m less than or equal to 5 has frequency 20; 5 less than m less than or equal to 15 has frequency 45; 15 less than m less than or equal to 40 has frequency 35. Calculate the frequency density for each class and state which class has the tallest bar.
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In a histogram of ages, the class 20 less than a less than or equal to 50 has frequency density 0.8. Estimate the number of people aged between 30 and 50, and state the assumption you have made.
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Free printable worksheet
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