8.2 Clustered and Stacked Bar Charts, Pie Charts, and Dual-Axis Line Graphs

Visual displays such as bar charts, pie charts, and time-series line graphs appear throughout school annual reports and national education datasets. While single-series bar charts are straightforward, LANTITE numeracy items frequently feature multi-segment stacked bars, sector angle conversions, dual vertical axes, and truncated baselines designed to test careful visual literacy.

Clustered, Stacked, and $100\%$ Stacked Bar Charts

Bar and column charts compare categorical groups across one or more sub-categories using three main layouts:

1. Clustered (side-by-side) bar/column charts: Each sub-category bar rests directly on the zero baseline ($0$). You read the value of every bar straight across to the axis scale.
2. Stacked bar/column charts (raw counts or dollars): Sub-categories for a single group are stacked on top of one another to form one composite bar whose total height equals the group total. Only the bottom segment starts at $0$. To find the value of any middle or top segment, you must subtract its bottom boundary from its top boundary:

$$\text{Segment Value} = \text{Top Edge Reading} - \text{Bottom Edge Reading}$$

Never read the top edge of a middle or upper segment as the value of that segment alone—the top edge gives the cumulative sum of that segment plus every segment beneath it! 3. $100\%$ stacked bar charts: Every composite bar is scaled to the exact same height ($100\%$), showing each sub-category's percentage share within its group.

Worked Example 1: Reading a Stacked Column Chart and a $100\%$ Stacked Chart

Suppose a stacked column chart tracks a school's camp funding in Term 2 across three stacked segments from bottom to top: School Grant (bottom), Parent Payments (middle), and Fundraising (top):

• The top of School Grant sits at $1,400 $\rightarrow$ Segment value is $1,400 minus $0, which equals $1,400.
• The top of Parent Payments sits at $3,650 $\rightarrow$ Segment value is $3,650 minus $1,400, which equals $2,250 (not $3,650!).
• The top of Fundraising sits at $4,500 $\rightarrow$ Segment value is $4,500 minus $3,650, which equals $850.

Now consider the classic $100\%$ stacked bar trap. Suppose a $100\%$ stacked chart shows that $35\%$ of students at North Campus walk to school, whereas only $20\%$ of students at South Campus walk to school. Which campus has more students walking to school?

• You cannot tell from the $100\%$ stacked chart alone without knowing the total enrolment of each campus!
• If North Campus has $200$ students, its $35\%$ segment represents $0.35 \times 200 = 70$ walkers.
• If South Campus has $500$ students, its shorter $20\%$ segment represents $0.20 \times 500 = 100$ walkers—which is $30$ more students than North Campus despite looking much smaller on the $100\%$ stacked chart.

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Pie (Sector) Charts: Angles, Percentages, and Counts

A pie chart partitions a full circle of $360^\circ$ into sectors whose central angles are proportional to each category's share of the total count $N$:

$$\frac{\theta}{360^\circ} = \frac{p\%}{100\%} = \frac{\text{Category Count}}{\text{Total Count } N}$$

Because $\frac{360^\circ}{100\%} = 3.6^\circ\text{ per }1\%$, you can convert rapidly between percentages, fractions, and sector angles using standard benchmarks:

Worked Example 2: Sector Angle to Student Count

In a commute survey of $N = 500$ students (shown in the bar chart below), $175$ students travel by Bus and $125$ travel by Car:

• Bus sector angle: $\frac{175}{500} \times 360^\circ = 0.35 \times 360^\circ = 126^\circ$ (or $35 \times 3.6^\circ = 126^\circ$).
• Car sector angle: $\frac{125}{500} \times 360^\circ = 0.25 \times 360^\circ = 90^\circ$.
• Conversely, if a sector has an angle of $32.4^\circ$ in a cohort of $500$ students, its percentage share is $\frac{32.4^\circ}{3.6^\circ} = 9\%$, which equals $0.09 \times 500 = 45$ students.

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Time-Series Line Graphs, Dual Axes, and Truncated Scales

A time-series line graph plots time intervals along the horizontal axis and a quantitative variable along the vertical axis:

• Rate of change (slope): The average rate of change between time $t_1$ and time $t_2$ is $\frac{y_2 - y_1}{t_2 - t_1}$. Visually, the steepest upward line segment represents the fastest rate of increase, while the steepest downward segment represents the fastest rate of decrease.
• Horizontal (flat) segments: A flat horizontal line between two time points means zero change (the value stayed constant at its previous level)—do not confuse a flat segment on a total enrolment graph with zero enrolments!
• Dual-axis (two vertical axes) graphs: Many educational charts plot two series with different units on the same canvas—for example, a column series for Total Enrolments (students) read against the left vertical axis, and a line series for Student-to-Teacher Ratio read against the right vertical axis. Always trace horizontally to the axis specified in the legend. Furthermore, the point where two lines cross visually on a dual-axis graph does not mean the two series have equal numerical values, because their left and right scales are independent.
• Truncated vertical axes (broken baseline): When a vertical axis starts at a non-zero number (such as $80\%$ instead of $0\%$), visual bar heights are distorted. On a chart whose vertical axis starts at $80\%$, a school with $88\%$ attendance has a bar $88 - 80 = 8$ units tall, whereas a school with $82\%$ attendance has a bar $82 - 80 = 2$ units tall—making $88\%$ appear four times as high as $82\%$ even though the actual difference is only $6$ percentage points ($88\% - 82\%$, or a relative ratio of $\frac{88}{82} \approx 1.073$).

Worked Example 3: Interpreting a Dual-Axis Enrolment and Staffing Graph

Suppose a dual-axis chart tracks a growing primary school from 2022 to 2025. The bars show Total Enrolments against the left axis ($0$ to $600$ students), while the plotted line shows the Student-to-Teacher Ratio against the right axis ($10.0$ to $20.0$ students per teacher). In 2024, the bar height reaches $480$ on the left axis and the line point sits at $16.0$ on the right axis. To find how many full-time-equivalent (FTE) teachers the school employed in 2024, combine the two axis readings:

$$\text{FTE Teachers} = \frac{\text{Total Enrolments (Left Axis)}}{\text{Student-to-Teacher Ratio (Right Axis)}} = \frac{480}{16.0} = 30\text{ teachers}$$