Grade 10 · Core Mathematics · Lesson 11 of 16

Advanced Data Representation: Histograms and Frequency Polygons

A Grade 10 Core Mathematics CBE lesson from the sub-strand Statistics I. In it, learners determine missing frequencies in grouped data and construct accurate histograms over continuous class boundaries.

Strand Statistics and ProbabilitySub-strand Statistics ILesson code 10CM-31-L1140 minutes · 3 slides

KICD learning outcome (d)

Represent data using histograms and frequency polygons.

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

Determine missing frequencies in grouped data and construct accurate histograms over continuous class boundaries.
Calculate class midpoints and construct closed frequency polygons including zero-frequency boundary anchors.
Draw combined histograms and frequency polygons and estimate median values from grouped statistical distributions.

Key vocabulary

Histogram: A graph of adjacent rectangular bars representing frequency distribution of grouped continuous data.; We draw a histogram to present the age distribution of factory workers.
Class boundary: The exact true limit separating consecutive grouped data classes without leaving gaps.; The class boundary between intervals $32 - 36$ and $37 - 41$ is $36.5$ .
Class midpoint: The exact middle numerical value of a specified class interval in statistics.; The class midpoint for motorcycle speeds in interval $40 - 49$ is $44.5$ .
Frequency polygon: A closed line graph formed by connecting midpoints of consecutive class intervals.; We plot frequency polygon vertices at class midpoints to display speed trends.
Linear interpolation: A mathematical estimation method assuming steady uniform change between known boundary values.; We use linear interpolation on grouped test marks to estimate the median.

Lesson preview

Slide 1 · Constructing Histograms and Determining Unknown FrequenciesModel · 12 min

Figure 1: A histogram displaying the frequency of examination marks using touching rectangular bars on continuous class boundaries.

To construct a histogram for grouped discrete or continuous data, such as examination marks, factory worker ages, or maize plant heights, convert class limits into continuous class boundaries. Adjacent rectangular bars touch without gaps. If total frequency is known, determine any missing class frequency $x$ first. The vertical axis represents frequency, while the horizontal axis displays continuous class boundaries. Use Figure 1 to see how touching bars are drawn on continuous class boundaries.

Histograms display grouped quantitative data using vertical touching rectangles constructed over exact class boundaries.
When total frequency is given, missing class frequencies are found by subtracting known frequencies from the total.
Equal class intervals produce histogram bars whose rectangular heights directly reflect the corresponding class frequencies.
Data such as factory worker ages, maize plant heights, or examination marks use continuous boundary columns.

Worked example. Sum the known candidate frequencies:

3 + 5 + 18 + 12 + 4 = 42

. Subtract from the total number of candidates

N = 50

x = 50 - 42 = 8

. Convert discrete class limits into continuous class boundaries by adjusting boundaries by 0.5: 9.5–19.5, 19.5–29.5, 29.5–39.5, 39.5–49.5, 49.5–59.5, 59.5–69.5. Draw rectangular bars of equal width (10 units) over these boundaries with heights matching frequencies: 3, 5, 8, 18, 12, 4. Answer: $x = 8$ ; Class boundaries are 9.5–19.5, 19.5–29.5, 29.5–39.5, 39.5–49.5, 49.5–59.5, and 59.5–69.5..

Check question: A factory records the ages of 40 workers in years across five classes: 30–34 (6), 35–39 (11), 40–44 ( $y$ ), 45–49 (9), 50–54 (4). Calculate $y$ and state the class boundaries for the class containing $y$ .

$y = 10$ ; boundaries 39.5 to 44.5
$y = 10$ ; boundaries 40 to 44
$y = 10$ ; boundaries 39.5 to 45.5
$y = 14$ ; boundaries 39.5 to 44.5

Show answer

$y = 10$ ; boundaries 39.5 to 44.5. Sum of known frequencies $= 6 + 11 + 9 + 4 = 30$ . Total workers $= 40$ , so $y = 40 - 30 = 10$ . The class interval 40–44 has continuous lower boundary $40 - 0.5 = 39.5$ and upper boundary $44 + 0.5 = 44.5$ .

Slide 2 · Constructing Frequency Polygons using Class MidpointsPractice · 14 min

Figure 2: A frequency polygon showing motorcycle speeds, constructed by plotting frequencies against class midpoints and connecting them with straight lines.

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

See the school projector WhatsApp us