8.4 Scatterplots, Trend Lines, Bubble Charts, and Correlation vs. Causation

Educational researchers and school improvement teams frequently examine how two numerical variables vary together—such as Attendance Rate vs. Assessment Score, Pre-Test vs. Post-Test performance, or Socio-Educational Advantage (ICSEA) vs. Numeracy Growth. A scatterplot displays such bivariate (two-variable) numerical data on a Cartesian coordinate plane.

Reading Scatterplots and Threshold Quadrants

In a scatterplot, each plotted point $(x_i, y_i)$ represents one individual observation (one student, one classroom, or one school) with two measurements:

• Horizontal ($x$) axis: Typically represents the explanatory (independent) variable or the earlier measurement (e.g., Term 1 Pre-Test Score or Attendance Percentage).
• Vertical ($y$) axis: Typically represents the response (dependent) variable or the later measurement (e.g., Term 4 Post-Test Score or Exam Mark).

Counting Points with Threshold Lines

LANTITE questions regularly ask how many students satisfy two criteria at once—for example, "How many students had an attendance rate below $85\%$ AND an assessment score above $70\%$?"

1. Draw (or mentally trace) the vertical threshold line at $x = 85\%$. Points strictly to the left have attendance $< 85\%$; points to the right have attendance $> 85\%$.
2. Draw the horizontal threshold line at $y = 70\%$. Points strictly above the line have a score $> 70\%$; points below have a score $< 70\%$.
3. For an AND condition (intersection), count only the points in the single quadrant that satisfies both requirements simultaneously (here, the top-left quadrant where $x < 85\%$ and $y > 70\%$). Check carefully whether points lying directly on a boundary line are included ("at least $70\%$" includes $y = 70\%$; "strictly above $70\%$" excludes $y = 70\%$). For an OR condition (union), count every point that satisfies at least one of the two criteria without double-counting the overlap.

The Identity Line $y = x$ in Pre-Test vs. Post-Test Plots

When the horizontal axis shows a Pre-Test Score ($x$) and the vertical axis shows a Post-Test Score ($y$) on the same scale, drawing the diagonal reference line $y = x$ separates the cohort into three distinct groups:

• Points ABOVE the line $y = x$: Here $y > x$ ($\text{Post-Test} > \text{Pre-Test}$), meaning the student improved between the two assessments. The vertical distance $y_i - x_i$ above the line equals the exact mark gain.
• Points ON the line $y = x$: Here $y = x$ ($\text{Post-Test} = \text{Pre-Test}$), meaning the student's score did not change.
• Points BELOW the line $y = x$: Here $y < x$ ($\text{Post-Test} < \text{Pre-Test}$), meaning the student's score decreased.

Worked Example 1: Comparing Growth Against the Line $y = x$

Suppose $20$ Year 5 students sit a $40$-mark numeracy diagnostic in Term 1 ($x$-axis) and an equivalent $40$-mark test in Term 4 ($y$-axis). A student plotted at $(22, 31)$ lies $31 - 22 = 9$ units vertically above the line $y = x$, representing a $9$-mark improvement (or a $\frac{9}{22} \times 100\% \approx 40.9\%$ relative gain over their starting score). Conversely, a student plotted at $(29, 25)$ lies $4$ units below the line $y = x$, indicating a $4$-mark drop. To find the student with the greatest absolute improvement, look for the point with the largest vertical distance above the diagonal line $y = x$—not necessarily the highest point on the chart!

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Direction, Form, Strength, Outliers, and Trend Lines

When describing the overall pattern of a scatterplot, statisticians examine four features:

Using a Line of Best Fit (Trend Line): Interpolation vs. Extrapolation

A line of best fit (or linear regression trend line) summarises a linear relationship with the equation:

$$\hat{y} = mx + c$$

• Slope ($m$): The predicted change in $y$ for each $1$-unit increase in $x$.
• Vertical intercept ($c$): The predicted value of $y$ when $x = 0$.
• Vertical residual ($y - \hat{y}$): A student whose plotted point lies above the trend line achieved an actual score $y$ higher than the score $\hat{y}$ predicted for their $x$-value; a point below the trend line scored lower than predicted.
• Interpolation (Reliable): Substituting an $x$-value within the observed range of $x$-data (for example, predicting the score for $x = 7$ hours of study when the dataset spans $2 \le x \le 12$ hours).
• Extrapolation (Unreliable): Extending the line to predict $y$ for an $x$-value outside the observed range (for example, plugging $x = 25$ hours into $\hat{y} = 4.5x + 38$ to predict $\hat{y} = 150.5$ out of $100$). Linear trends rarely hold indefinitely outside the observed domain.

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Multi-Variable Bubble Charts and Correlation vs. Causation

Reading 3-Variable and 4-Variable Bubble Charts

A bubble chart enhances a standard 2D scatterplot by encoding one or two additional variables into the plotted circles:

1. $x$-axis position: Variable 1 (e.g., School ICSEA index).
2. $y$-axis position: Variable 2 (e.g., Mean Year 9 Numeracy score).
3. Bubble size (area or diameter): Variable 3, a quantitative magnitude (e.g., Total school enrolment $N$). Larger bubbles represent larger schools.
4. Bubble colour or shading: Variable 4, typically a categorical classification (e.g., Metropolitan vs. Regional/Remote sector).

Correlation Does Not Imply Causation

Even when a scatterplot displays a strong positive or negative correlation between $X$ and $Y$, observational data alone never proves that $X$ causes $Y$. A third confounding (lurking) variable $Z$ often drives both variables simultaneously:

• Across Years 1 to 6 in a primary school, there is a strong positive correlation between student shoe size ($X$) and reading comprehension score ($Y$).
• Buying larger shoes does not cause a child to read better! Instead, chronological age / year level ($Z$) is a confounding variable that independently increases both physical foot size and reading proficiency.
• Similarly, a positive correlation between number of books in the home and NAPLAN reading achievement reflects broader socioeconomic and family literacy factors rather than a guarantee that simply shipping boxes of books to a house will automatically raise test scores by a fixed slope.