Representing Data Using Histograms and Frequency Polygons

By the end of this lesson, learners should be able to

1. Determine continuous class boundaries and uniform class widths from grouped frequency distributions.
2. Construct accurate histograms with adjacent touching bars representing class frequencies on a Cartesian plane.
3. Construct frequency polygons by plotting frequencies against class midpoints and anchoring both ends to the horizontal axis at zero-frequency points.

Key vocabulary

Histogram

A graph of grouped continuous data using touching rectangular bars over class boundaries.

We draw a histogram with touching bars to represent waiting times at a dispensary.

Class boundary

The exact value separating adjacent class intervals so that histogram bars meet without gaps.

The class interval 10 to 14 has lower boundary 9.5 and upper boundary 14.5.

Frequency

The number of observations or data values falling within a specific class interval.

A class interval with frequency 12 produces a histogram bar 12 units high.

Frequency polygon

A line graph joining class midpoints at frequencies, anchored to the x-axis at zero frequency.

We plot frequencies against midpoints to draw a frequency polygon anchored on the axis.

Midpoint

The central numerical value calculated as half the sum of a class interval's limits.

The midpoint of class interval 12 to 20 is $(12+20)\mathrm{/}2=16$.

Lesson preview

Slide 1 · Understanding Histograms and Class BoundariesOpening · 12 min

When continuous or grouped data is recorded in equal class intervals, we present it visually using a histogram. Unlike bar charts for discrete categories, a histogram consists of rectangular bars that touch each other without spaces. The horizontal base of each bar extends from the lower class boundary to the upper class boundary. When class widths are uniform, the height of each bar directly corresponds to the frequency of that class interval. Use Figure 1 to observe how the horizontal base of each bar extends between continuous class boundaries.

• A histogram represents grouped data using touching rectangular bars with no gaps between them.
• Class boundaries remove gaps between discrete class limits to form a continuous horizontal scale.
• For equal class widths, the height of each bar represents the frequency of the class.
• Lower and upper class boundaries determine the exact start and end of each rectangular bar.

Worked example. Find the boundary gap between adjacent classes: $15-14=1$, so subtract 0.5 from lower limits and add 0.5 to upper limits. Class 10–14: lower boundary = $10-0.5=9.5$, upper boundary = $14+0.5=14.5$. Width = $14.5-9.5=5$. Class 15–19: boundaries are 14.5 to 19.5, width = 5. Class 20–24: boundaries are 19.5 to 24.5, width = 5. Class 25–29: boundaries are 24.5 to 29.5, width = 5. Answer: Class boundaries are 9.5–14.5, 14.5–19.5, 19.5–24.5, and 24.5–29.5; uniform class width is 5 units..

Check question: A grouped frequency distribution has class intervals 5–9, 10–14, and 15–19. What is the upper class boundary of the first class and the lower class boundary of the second class?

1. Both are 9.5
2. Upper is 9.0, lower is 10.0
3. Both are 10.0
4. Both are 9.0

Show answer

Both are 9.5. Subtracting 0.5 from the lower limit 10 gives 9.5, and adding 0.5 to the upper limit 9 gives 9.5, ensuring adjacent bars touch at exactly $x=9.5$.

Slide 2 · Constructing Histograms from Grouped Frequency TablesModel · 14 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 15 lessons in this sub-strand

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All 16 lessons in Statistics I

1. 1Gathering Data from Practical SituationsCollect data from real-life sources
2. 2Frequency Distribution Tables for Ungrouped and Grouped DataDraw a frequency distribution table for grouped and ungrouped data
3. 3Drawing Grouped Frequency Distribution TablesDraw a frequency distribution table for grouped and ungrouped data
4. 4Constructing Frequency Distribution Tables for Grouped and Ungrouped DataDraw a frequency distribution table for grouped and ungrouped data
5. 5Calculating Averages and Middle Values of Raw DataDetermine mean, mode and median of grouped and ungrouped data
6. 6Determining Median of Data and Introduction to Grouped Data MeanDetermine mean, mode and median of grouped and ungrouped data
7. 7Determining Mean, Modal Class, and Median of Grouped DataDetermine mean, mode and median of grouped and ungrouped data
8. 8Determining Mean, Mode, and Median of Grouped and Ungrouped DataDetermine mean, mode and median of grouped and ungrouped data
9. 9Representing Data Using Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
10. 10Constructing Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
11. 11Advanced Data Representation: Histograms and Frequency PolygonsRepresent data using histograms and frequency polygons
12. 12Interpreting Data from Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
13. 13Interpreting Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
14. 14Interpreting Histograms and Frequency PolygonsInterpret data from histograms and frequency polygons
15. 15Data Collection, Organisation, and Informed Decision MakingPromote data collection, organisation and representation for informed decision making
16. 16Data Interpretation and Decision MakingPromote data collection, organisation and representation for informed decision making