Bar charts, pie charts, histograms and frequency polygons – Descriptive Statistics (Senior 4 Basic Mathematics) | Ganzaa

Descriptive Statistics

Lessons

Senior 4 · Basic Mathematics

Descriptive Statistics

5 lessons

Lesson 2 of 5

Bar charts, pie charts, histograms and frequency polygons

25 min

Which picture answers which question?

The 2022 census gives each province's population: City of Kigali 1 745 555, Southern 3 002 699, Western 2 896 484, Northern 2 038 511 and Eastern 3 563 145 (NISR, Table 1). A planner may want to compare the provinces with each other, or to show each province's share of the whole country. A geographer may want the shape of the 30 district populations from the last lesson. Each question needs a different graph, and choosing the wrong one can hide the answer or even mislead.

Definition

Bar chart: separate bars of equal width whose heights show the frequencies of categories or discrete values; gaps between bars. Pie chart: a circle divided into sectors whose angles are proportional to the frequencies:

\text{sector angle} = \frac{f}{\sum f} \times 360^\circ

Histogram: touching bars over the class boundaries of grouped continuous (or treated-as-continuous) data, whose areas are proportional to the frequencies. Frequency density $= \frac{\text{frequency}}{\text{class width}}$ . Frequency polygon: the midpoints of the tops of the histogram bars joined by straight lines.

Bar charts and pie charts compare categories

The bar chart makes comparisons easy: the Eastern Province is the most populous, the City of Kigali the least, and the Southern and Western Provinces are almost equal. A pie chart shows the same data as shares of the whole.

Histograms and frequency polygons show a distribution

For grouped numerical data, the bars of a histogram touch, because the classes run on without gaps. When all classes have the same width, bar heights can show the frequencies directly.

The shape is skewed to the right (positively skewed): most districts are bunched at the low end, and a long thin tail stretches to the right, ending with Gasabo. A frequency polygon shows the same shape as a line, and is useful for comparing two distributions on one grid. It is closed by joining it to the horizontal axis at the midpoints of the empty classes on each side (250 and 950).

Tip

Choose by the question. Compare categories: bar chart. Show parts of a whole: pie chart (five or six sectors at most). Show the shape of grouped numerical data: histogram. Compare two distributions: frequency polygons on one grid. And always give a title, labelled axes with units, and the source.

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Worked example 1: sector angles

Calculate the sector angle for each province for a pie chart of the 2022 population (total 13 246 394).

\text{Kigali: } \frac{1\,745\,555}{13\,246\,394} \times 360^\circ \approx 47.4^\circ, \qquad \text{Eastern: } \frac{3\,563\,145}{13\,246\,394} \times 360^\circ \approx 96.8^\circ

Similarly Southern $\approx 81.6^\circ$ , Western $\approx 78.7^\circ$ and Northern $\approx 55.4^\circ$ . Check: $47.4 + 81.6 + 78.7 + 55.4 + 96.8 = 359.9^\circ$ ; the missing $0.1^\circ$ is rounding, so the angles are consistent with $360^\circ$ .

Worked example 2: unequal class widths

A report groups the 30 districts as $300 \le x \lt 400$ (15 districts), $400 \le x \lt 500$ (10) and $500 \le x \lt 900$ (5), in thousands. Draw a correct histogram.

The last class is four times as wide. Measure widths in units of 100 thousand and divide:

\frac{15}{1} = 15, \qquad \frac{10}{1} = 10, \qquad \frac{5}{4} = 1.25\ \text{districts per 100 thousand}

Why it matters: a bar of height 5 over the wide class would have an area of $5 \times 4 = 20$ units, larger than the 400–500 bar (10 units), suggesting more districts above 500 000 than between 400 000 and 500 000, which is false. With frequency density, area $= 1.25 \times 4 = 5$ : correct. The frequency of any class can be read back as density × width.

Worked example 3: interpreting a graph

Use the histogram of district populations and the density map to write three statements a planner could use.

Half of all districts (15 of 30) have 300 000 to 400 000 residents, so services planned for a "typical" district should fit about that size.
The distribution is skewed to the right: one district (Gasabo, 879 505) has more than twice the population of most districts, and needs services on a different scale.
The map shows that the most crowded sectors are urban (Kigali and secondary cities), while many eastern sectors have fewer than 250 people per km², so the same number of schools must cover longer distances there.

Each statement quotes a figure from the graph and says what it means: that is what "interpret" requires.

Warning

Misleading graphs. A bar chart whose vertical axis starts at 1.5 million instead of 0 makes Kigali look tiny next to the other provinces. A histogram with unequal classes drawn with frequency, not density, distorts the shape. A pie chart with too many sectors, or drawn in 3D, makes angles hard to compare. When you read a graph, check the axis scales, the class widths and the source before you believe it.

Corbettmaths, "Reading Histograms". Watch how frequency = frequency density × class width, then use it to read back the 5 districts in the wide class of worked example 2.

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Practice

Try each one before opening the answers. They get harder as you go.

A family's monthly spending of 240 000 Frw is: food 108 000, rent 60 000, transport 36 000, school 24 000, other 12 000 (illustrative). Calculate the sector angles for a pie chart.
From the bar chart, by how many people did the Eastern Province exceed the Northern Province in 2022? What percentage of the national population is that?
The times (minutes) 40 pupils take to walk to school are: $0 \le t \lt 10$ : 6; $10 \le t \lt 20$ : 14; $20 \le t \lt 30$ : 12; $30 \le t \lt 60$ : 8 (illustrative). Calculate the frequency density of each class, per minute.
In question 3, which class has the tallest bar in a correct histogram? Explain why the 30–60 class, despite its frequency of 8, has a short bar.
A histogram has a bar over $20 \le x \lt 50$ with frequency density 0.4 per unit. What is the frequency of that class?
Give two reasons why a pie chart would be a poor choice for the 30 district populations.
A newspaper chart shows a "huge rise" in bus fares, but its vertical axis runs from 450 Frw to 500 Frw. The fare rose from 460 Frw to 490 Frw. Calculate the real percentage rise and explain how the chart misleads.

Show answersHide answers

Food $\frac{108}{240} \times 360^\circ = 162^\circ$ ; rent $90^\circ$ ; transport $54^\circ$ ; school $36^\circ$ ; other $18^\circ$ . Check: total $360^\circ$ .
$3\,563\,145 - 2\,038\,511 = 1\,524\,634$ people; $\frac{1\,524\,634}{13\,246\,394} \times 100\% \approx 11.5\%$ of the population.
$\frac{6}{10} = 0.6$ , $\frac{14}{10} = 1.4$ , $\frac{12}{10} = 1.2$ , $\frac{8}{30} \approx 0.27$ pupils per minute.
The 10–20 class (density 1.4). The 30–60 class is three times as wide, so its 8 pupils are spread over 30 minutes: density only 0.27; its area $0.27 \times 30 = 8$ still shows the right frequency.
$f = 0.4 \times (50 - 20) = 12$ .
Thirty sectors are far too many to read; the angles of districts of 360 000 and 370 000 would look identical; a pie chart shows shares, not the shape of the distribution, which is what the data's interest lies in.
$\frac{490 - 460}{460} \times 100\% \approx 6.5\%$ . Because the axis starts at 450, the bars appear as heights 10 and 40, four times as tall, suggesting a 300% rise.

Quick check

Key points

Bar charts compare categories (heights); pie charts show shares (angle $= \frac{f}{\sum f} \times 360^\circ$ ).
Histograms show grouped numerical data: touching bars, area proportional to frequency.
With unequal classes, plot frequency density $= \frac{f}{\text{class width}}$ ; frequency = density × width.
A frequency polygon joins the midpoints of the bar tops; it is good for comparing distributions.
Rwanda's district populations are skewed to the right. Check axes, widths and sources before trusting any graph.

Sources and credits

Data: Fifth Rwanda Population and Housing Census, 2022: Main Indicators Report, National Institute of Statistics of Rwanda (Table 1).
Map: Rwanda populationdensity sector 2022, Aquietfarwell, Wikimedia Commons, CC BY-SA 4.0.
Video: Reading Histograms, Corbettmaths, embedded from YouTube.
Written by Ganzaa: the explanations, the four charts (drawn from NISR data), the worked examples, the practice items and the questions. The family budget, walking times and bus fares are illustrative. This lesson uses CC BY-SA material and is shared under CC BY-SA 4.0.

1Which picture answers which question?

2Bar charts and pie charts compare categories

3Histograms and frequency polygons show a distribution

Which picture answers which question?

Bar charts and pie charts compare categories

Histograms and frequency polygons show a distribution