Subsection 3.1.4 Histograms and Frequency Polygons

Curriculum Alignment

3.0 Statistics and Probability

Sub-Strand

3.1 Statistics 1

Specific Learning Outcomes

Represent data using histograms and frequency polygons
Interpret data from histograms and frequency polygons

Teacher Resource 3.1.103.

To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.

Lesson Plan

Step-by-Step Guide

Representation of data is the process of presenting collected information (data) in an organized and visual form so that it is easy to understand, interpret and analyze.

Instead of leaving data as raw numbers, we use tables, charts and graphs to show patterns, trends, and comparisons more clearly.

Learner Experience 3.1.104.

Work in groups

In a school with \(500\) students, their heights were measured and recorded in the following table.

Table 3.1.105.

Height (cm)	Number of Students (Frequency)
\(140 - 149\)	\(30\)
\(150 - 159\)	\(70\)
\(160 - 169\)	\(110\)
\(170 - 179\)	\(150\)
\(180 - 189\)	\(90\)
\(190 - 199\)	\(50\)

Choose a suitable scale and represent the data on a histogram and a frequency polygon.
Compare and discuss your graphs with other groups.

Key Takeaway 3.1.106.

A histogram uses adjacent bars to show frequency distribution, while a frequency polygon connects the midpoints of the bars with a line to show patterns.

\(\textbf{Class Width}\) is the difference between the upper and lower boundaries of a class.

\(\textbf{Equal class width}\) means all bars have the same width.

\(\textbf{Unequal class width}\) means bars have different widths to better represent uneven data.

\(\textbf{Frequency density}\) is a measure used in histograms to ensure that the area of each bar represents the actual frequency of observations, especially when class widths are unequal.

Frequency density is calculated using the formula:

\begin{align*} \textbf{Frequency density} = \amp \frac{\textbf{Frequency}}{\textbf{Class width}} \end{align*}

Where;

\(\textbf{Frequency}\) is the number of observations in a class interval.

\(\textbf{Why Use Frequency Density Instead of Frequency?}\)

In a histogram, the area of each bar (not just the height) represents the frequency.
If class widths are unequal, simply plotting frequency would distort the representation.
Using frequency density ensures that the area of each bar remains proportional to the actual frequency.

\(\textbf{Midpoint}\) of a class interval represents the central value of that range. It is the average of the lower and upper boundaries of the class.

Formula for midpoint:

\begin{align*} \textbf{Midpoint} = \amp \frac{\textbf{Lower bound + Upper bound}}{2} \end{align*}

Example 3.1.107.

The table below presents the salary distribution of employees in a company.

Table 3.1.108.

Salary Range (KSh)	Frequency
\(1000 - 1500\)	\(42\)
\(1500 - 2000\)	\(35\)
\(2000 - 2500\)	\(20\)
\(2500 - 3000\)	\(15\)
\(3000 - 4000\)	\(18\)
\(4000 - 5000\)	\(42\)

Draw a histogram and a frequency polygon to represent the data.

Solution.

To draw a histogram and a frequency polygon, we need to find the frequency density and the midpoint of each class interval.

We use frequency density instead of frequency to draw the histogram because the class widths are unequal.

The formula for frequency density:

\begin{align*} \textbf{Frequency density} = \amp \frac{\textbf{Frequency}}{\textbf{Class width}} \end{align*}

The formula for Midpoint:

\begin{align*} \textbf{Midpoint} = \amp \frac{\textbf{Lower bound + Upper bound}}{2} \end{align*}

Table 3.1.109.

Salary range(Ksh.)	Frequency	Class width	Frequency density	Midpoint
\(1000 - 1500\)	\(42\)	\(500\)	\(0.084\)	\(1250\)
\(1500 - 2000\)	\(35\)	\(500\)	\(0.070\)	\(1750\)
\(2000 - 2500\)	\(20\)	\(500\)	\(0.040\)	\(2250\)
\(2500 - 3000\)	\(15\)	\(500\)	\(0.030\)	\(2750\)
\(3000 - 4000\)	\(18\)	\(1000\)	\(0.018\)	\(3500\)
\(4000 - 5000\)	\(42\)	\(1000\)	\(0.042\)	\(4500\)

Example 3.1.110.

The following frequency distribution shows the daily rainfall amounts (in mm) recorded at a weather station over a \(60\) day period.

Table 3.1.111.

Rainfall(mm)	Frequency
\(0 - 5\)	\(22\)
\(6 - 10\)	\(15\)
\(11 - 15\)	\(12\)
\(16 - 25\)	\(8\)
\(26 - 40\)	\(3\)

Create a histogram to represent this data.

Solution.

To draw a histogram we need to find the frequency density because class width are unequal.

The formula for frequency density:

\begin{align*} \textbf{Frequency density} = \amp \frac{\textbf{Frequency}}{\textbf{Class width}} \end{align*}

Table 3.1.112.

Rainfall(mm)	Frequency	Class width	Frequency density
\(5 - 10\)	\(22\)	\(5\)	\(4.4\)
\(10 - 15\)	\(15\)	\(5\)	\(3.0\)
\(15 - 20\)	\(12\)	\(5\)	\(2.4\)
\(20 - 30\)	\(8\)	\(10\)	\(0.8\)
\(30 - 45\)	\(3\)	\(15\)	\(0.2\)

Checkpoint 3.1.113.

Checkpoint 3.1.114.

Checkpoint 3.1.115.

Exercises Exercises

1.

The following data represents the heights (in cm) of \(30\) students in a class:

\(150, 155, 160, 162, 165, 158, 170, 172, 168, 153, 163, 167, 175, 178, 161,\)

\(156, 169, 171, 159, 164, 173, 176, 157, 166, 174, 177, 154, 165, 179, 160\)

Create a frequency table with class intervals of \(5 \textbf{ cm}\) and midpoints of the data.
Using the frequency table you created Draw a histogram to represent the data.
Draw a frequency polygon on the same axes as your histogram.
Label your axes and give your graphs appropriate titles.

Answer.

Table 3.1.116. Heights of 30 students

Height (cm) Frequency Midpoint

150–154 3 152

155–159 5 157

160–164 6 162

165–169 6 167

170–174 5 172

175–179 5 177

Height (cm)	Frequency	Midpoint
150–154	3	152
155–159	5	157
160–164	6	162
165–169	6	167
170–174	5	172
175–179	5	177

2.

The following data represents the ages of \(25\) people in a community meeting:

\(20, 25, 30, 35, 40, 22, 28, 33, 38, 42, 27, 32, 37, 41,\)

\(24, 29, 34, 39, 43, 26, 31, 36, 44, 23, 45\)

Create a frequency table with class intervals of \(5\) years
Draw a histogram to represent the data.
Draw a frequency polygon on the same axes.
Label the axes and provide titles for your graphs.

Answer.

Table 3.1.117. Ages of 25 people

Age (years) Frequency

20–24 4

25–29 5

30–34 5

35–39 5

40–44 5

45–49 1
Histogram and frequency polygon

Age (years)	Frequency
20–24	4
25–29	5
30–34	5
35–39	5
40–44	5
45–49	1

3.

A school collected data on the number of books read by students in a term. The following frequency table shows the results:

Table 3.1.118.

Number of Books Read	Frequency
\(0 - 2\)	\(15\)
\(3 - 5\)	\(25\)
\(6 - 8\)	\(35\)
\(9 - 11\)	\(15\)
\(12 - 14\)	\(10\)

Draw a histogram to represent the data from your frequency table.
On the same axes, draw a frequency polygon.
Estimate the median number of books read. Explain your reasoning.

Answer.

Histogram and frequency polygon
Estimated median number of books

Median \(\approx 7\) books

4.

A survey was conducted to find out how much time people spend on social media daily. The following data was collected:

Table 3.1.119.

Time (Minutes)	Frequency	Class Width	Frequency Density
\(0 - 10\)	\(15\)	\(10\)	\(1.5\)
\(10 - 20\)	\(25\)	\(10\)	\(2.5\)
\(20 - 30\)	\(30\)	\(10\)	\(3.0\)
\(30 - 60\)	\(40\)	\(30\)	\(1.33\)
\(60 - 120\)	\(20\)	\(60\)	\(0.33\)

Draw a histogram to represent the sales data.
Draw a frequency polygon to represent the sales data.
Label your axes and provide appropriate titles for your graphs.

Answer.

(a), (b) Histogram and frequency polygon (frequency density)

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