For numerical wing length and numerical descent time, a scatter plot can show the relationship, with wing length on the horizontal axis and time on the vertical axis. For unordered material categories, separate bars or points may be clearer. A pie chart generally does not show how a measured response changes with a numerical independent variable.
Label each axis with the quantity and unit. Use a consistent numerical scale and display the observed range honestly. You may draw a justified trend or fit, but do not connect unrelated categories as if intermediate values exist. Retain the table so readers can inspect the exact records.
Book notes: Choose the representation for the question
For numerical wing length and numerical descent time, a scatter plot can show the relationship, with wing length on the horizontal axis and time on the vertical axis. For unordered material categories, separate bars or points may be clearer. A pie chart generally does not show how a measured response changes with a numerical independent variable.
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Which graph is suitable for numerical wing length versus descent time?
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Show the size of change and variability
Show the size of change and variability
A truncated vertical axis can make a small change look large. A non-zero starting axis is not automatically wrong for a scatter plot, but it must be clearly labelled and interpreted using numerical values rather than visual drama. Use zero appropriately for bar-length comparisons and disclose any break in scale.
If plotting error bars, state exactly what they represent: range, standard deviation, instrument uncertainty or another interval. These are different quantities. Overlap alone is not a universal statistical test, and the absence of overlap does not establish that an uncontrolled factor cannot explain the difference.
Book notes: Show the size of change and variability
A truncated vertical axis can make a small change look large. A non-zero starting axis is not automatically wrong for a scatter plot, but it must be clearly labelled and interpreted using numerical values rather than visual drama. Use zero appropriately for bar-length comparisons and disclose any break in scale.
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Which value is an interpolation using lengths 3, 5 and 7 cm?
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Restrict predictions to the evidence
Restrict predictions to the evidence
A pattern in three wing lengths may support an estimate between those lengths, if the relationship is reasonably smooth. It does not establish the same trend for extremely large wings, where folding, stability or material behaviour may change. State that an interpolation is an estimate, not a measured value.
Look for outliers, scatter and curvature before selecting a model. A line fitted to observations is a representation, not a new cause. A graph helps communicate evidence, but interpretation still depends on the method, provenance and limitations.
Book notes: Restrict predictions to the evidence
A pattern in three wing lengths may support an estimate between those lengths, if the relationship is reasonably smooth. It does not establish the same trend for extremely large wings, where folding, stability or material behaviour may change. State that an interpolation is an estimate, not a measured value.
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A synthetic spinner dataset gives mean times of 1.3 s at 3 cm and 1.7 s at 5 cm. Using a local straight-line pattern, what is the estimated mean time at 4 cm?
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Worked example
Worked example: Estimate mean time at 4 cm from the synthetic table, then judge a prediction at 30 cm.
4 cm lies between the supplied modelled lengths of 3 and 5 cm.
A straight-line interpolation gives the midpoint of 1.3 and 1.7 s: 1.5 s.
Label 1.5 s as an estimate conditional on that local pattern.
30 cm is outside the observed range; stability and geometry may change, so the same line is not established there.
4 cm: estimated 1.5 s by local linear interpolation. A 30 cm prediction is unvalidated extrapolation.
Synthetic teaching dataset: mean spinner times; bars, if drawn, would represent the observed trial range.
Wing length (cm)
Mean descent time (s)
Trial minimum (s)
Trial maximum (s)
3
1.3
1.2
1.4
5
1.7
1.6
1.8
7
2.1
2
2.2
Read a falling-object distance–time graph
Synthetic idealised falling-object data, generated using distance = 5 × time². Assumptions: release from rest, downward distance from release, constant acceleration 10 m/s² and negligible air resistance. Not measured data.Text description
Time in seconds is on the horizontal axis and downward distance in metres on the vertical axis. Points are (0, 0), (0.1, 0.05), (0.2, 0.20), (0.3, 0.45) and (0.4, 0.80). The plotted points rise increasingly steeply. Straight segments join the sampled points as a visual guide, not exact motion between samples.
Time in seconds is on the horizontal axis and downward distance in metres on the vertical axis. Points are (0, 0), (0.1, 0.05), (0.2, 0.20), (0.3, 0.45) and (0.4, 0.80). The plotted points rise increasingly steeply. Straight segments join the sampled points as a visual guide, not exact motion between samples.
Synthetic sampled downward distance
Time since release (s)
Downward distance (m)
0
0
0.1
0.05
0.2
0.2
0.3
0.45
0.4
0.8
Compare the distance travelled in 0.0–0.1 s with 0.3–0.4 s. Calculate average speed in each interval and describe the pattern.
First interval distance = 0.05 − 0 = 0.05 m; final interval distance = 0.80 − 0.45 = 0.35 m. Each interval lasts 0.10 s.
Average speed in the first interval = 0.05 ÷ 0.10 = 0.5 m/s; in the final interval = 0.35 ÷ 0.10 = 3.5 m/s.
More distance is covered in equal time intervals, so the model’s speed increases. These are interval-average speeds, not the instantaneous speed at an endpoint.
The graph shows the model assumptions, not proof that all real objects follow it. A real recording requires a calibrated distance scale, timing information and attention to air resistance.
The last interval covers seven times the distance of the first (0.35/0.05 = 7). Average speeds are 0.5 and 3.5 m/s. The increasingly steep graph represents speeding up, not a constant falling speed.
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Activities
Activities
Activity 1
Specify axes and a suitable graph type for the supplied spinner dataset, including the meaning of any bars.
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Activity 2
A graph uses 1.29 to 1.31 s on the vertical axis. Explain how to judge the magnitude of change.
Five questions from this lesson’s eight-question bank
Answer the five-question Quick Check. Use the feedback to repair an idea before retrying.
Short answer
Explain and evaluate
Q1. Describe a suitable graph for the supplied spinner table.
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Explain and evaluate
Q2. Distinguish the evidential status of the supplied synthetic 5 cm time, interpolated 4 cm time and extrapolated 30 cm time.
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Explain and evaluate
Q3. Explain why a steep-looking graph does not alone prove a large or causal effect.
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Answers and model responses
Activities
Activity 1
Scatter plot: x wing length (cm), y mean descent time (s). If bars show trial minimum-to-maximum, label them observed trial ranges rather than standard deviations.
Activity 2
Read numerical differences and units and note the restricted scale. Apparent height on the page does not establish a large physical effect.
Short Answer Model Answers
Q1: Use a scatter plot with wing length (cm) horizontally and mean descent time (s) vertically, a consistent labelled scale, and points at 3/5/7 cm. If ranges are shown, define them explicitly.
Q2: 5 cm is a supplied synthetic value; 4 cm is interpolation under a local model; 30 cm is extrapolation beyond the range. None are class empirical measurements in this dataset.
Q3: Axis scaling affects appearance. Read numerical magnitude and variability; causal interpretation also requires a valid design controlling relevant alternatives, not simply a trend.
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Revisit and reflect
Revisit without looking
Select a graph for the data structure.
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Check the key idea
4 cm: estimated 1.5 s by local linear interpolation. A 30 cm prediction is unvalidated extrapolation.