Extended. When classes have different widths, a bar's height cannot be its frequency. A histogram plots frequency density instead, so that the AREA of each bar is the frequency.
Frequency density \(= \dfrac{\text{frequency}}{\text{class width}}\). Plot frequency density on the vertical axis; then frequency \(=\) frequency density \(\times\) class width, which is the area of the bar.
Fig. 1: histogram of the masses of 80 parcels, drawn to scale (frequency densities 0.7, 1.5, 2.2, 1.2, 0.4). The shaded area from 25 kg to 40 kg is 5 × 2.2 + 10 × 1.2 = 23 parcels.
Section 01
Frequency density.
In the lab's histogram tab, tick 'show the mistake: height = frequency' and see how the wide 30 to 50 class towers over the others. Untick it: with frequency density on the axis, the wide bar becomes low and spread out, and its area shows its true frequency.
Class width = upper boundary \(-\) lower boundary: \(30 < m \le 50\) has width 20.
Frequency density = frequency \(\div\) class width. Label the vertical axis 'Frequency density'.
Bars touch (the data are continuous), and each bar spans its class exactly on the horizontal axis.
The area of each bar equals its frequency, so wide classes are drawn lower.
Worked example Draw a histogram
The masses, \(m\) kg, of 80 parcels are shown. Calculate the frequency densities for a histogram.
Draw bars from 0 to 10, 10 to 20, ..., 50 to 80 with these heights (Fig. 1).
Answer: 0.7, 1.5, 2.2, 1.2, 0.4
Examiner trap
The 30 to 50 class has the biggest frequency (24) but NOT the tallest bar: its density is only 1.2. Plotting frequency instead of frequency density is the mistake examiners look for first.
Section 02
Reading a histogram: area.
To get information out of a histogram, turn heights back into frequencies: frequency = frequency density \(\times\) class width. For part of a class, use the part of the width you need.
Worked example A frequency from a bar
In a histogram, the bar for \(40 < x \le 60\) has a frequency density of 1.6. Find the frequency.
Class width \(= 60 - 40 = 20\).
Frequency \(= 1.6 \times 20 = 32\).
Answer: 32
Worked example Estimate a count in a range
Use Fig. 1 to estimate the number of parcels with masses between 25 kg and 40 kg.
From 25 to 30 kg: half of the \(20 < m \le 30\) bar, width 5: \(5 \times 2.2 = 11\).
From 30 to 40 kg: part of the \(30 < m \le 50\) bar, width 10: \(10 \times 1.2 = 12\).
Total \(= 11 + 12 = 23\). It is an estimate because we assume the values are spread evenly within each class.
Answer: About 23 parcels
Your turn
Use Fig. 1 to estimate the number of parcels heavier than 60 kg.
60 to 80 kg is width 20 of the \(50 < m \le 80\) bar: \(20 \times 0.4 = 8\) parcels.
Examiner trap
Using the whole class when only part is wanted overcounts: 25 to 40 kg is NOT \(22 + 24 = 46\). Use only the widths 5 and 10 inside the range.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
How is frequency density calculated?
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The class \(10 < x \le 30\) has frequency 50. What is its frequency density?
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In a histogram, what does the area of a bar represent?
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How should the vertical axis of a histogram be labelled?
Section 03
When the scale is hidden.
Harder questions give a histogram with no numbers on the frequency density axis, plus the frequency of one class. Use that class to find what one unit of height is worth, then use areas for the rest.
Worked example Find the scale
In a histogram, the bar for \(10 < t \le 20\) represents 30 people and is 4.5 cm tall. The bar for \(20 < t \le 25\) is 6 cm tall. How many people does it represent?
Frequency density of the first bar \(= 30 \div 10 = 3\), shown by 4.5 cm, so 1 cm represents \(3 \div 4.5 = \tfrac{2}{3}\).
Second bar: frequency density \(= 6 \times \tfrac{2}{3} = 4\).
Frequency \(= 4 \times 5 = 20\) people.
Answer: 20 people
Examiner trap
Comparing bar heights alone gives \(\dfrac{6}{4.5} \times 30 = 40\), which ignores that the second class is only half as wide. Always multiply by the class width.
Stage 2 · Exam
Exam-style questions
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The class \(20 < h \le 35\) has frequency 45. Find the height of its bar in a histogram.
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A histogram bar for \(40 < x \le 60\) has frequency density 1.6. Find the frequency of this class.
Exam style+20 XP
Use Fig. 1 to estimate the number of parcels with masses between 25 kg and 40 kg.
Exam style+20 XP
In a histogram, the bar for \(10 < t \le 20\) represents 30 people and is 4.5 cm tall. The bar for \(20 < t \le 25\) is 6 cm tall. How many people does it represent?
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Histograms, by area.
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