Paper Calibration Curve: Predicting Leaf Area from Mass

The regression from Activity 11, written up as a report

Introduction

To estimate leaf surface area without a leaf-area meter, we built a calibration curve. Paper from one stock has a constant area per gram, so the mass of a paper square is a near-perfect linear predictor of its area. Once we know that relationship, we can trace a leaf onto the same paper, weigh the tracing, and convert its mass to an area.

• Explanatory variable (X): paper mass (g) — easy to measure on a balance
• Response variable (Y): paper area (cm²) — what we actually want

Methods

We weighed 118 paper squares of known area (1–567 cm²) and fit a simple linear regression, area_cm2 ~ mass_g, with lm(). We checked the residuals with base R’s diagnostic plots for the fitted model. All work was done in R with the tidyverse; this report was written in Quarto, so every number and figure below is computed from the data rather than typed in by hand.

Results

Paper mass was an extremely strong predictor of paper area. The overall fit is in Table 1 and the fitted equation is in Table 2.

Table 1. Overall fit of the calibration regression.

R-squared	F	df (model)	df (residual)	p-value
0.9999	784932.9	1	116	0

Table 2. Coefficients of the calibration equation. The mass_g estimate is the area gained per additional gram of paper; (Intercept) is the fitted area at zero mass.

Term	Estimate	Std. error	t	p-value
(Intercept)	0.279	0.181	1.5	0.1261386
mass_g	130.523	0.147	886.0	0.0000000

Read the equation off Table 2: predicted area equals the (Intercept) estimate plus the mass_g estimate times the mass. In words, each additional gram of this paper stock adds about the mass_g estimate in cm² of area. The slope’s p-value tests whether mass predicts area at all; the intercept is not distinguishable from zero, which is what physics says it should be — paper with no mass has no area.

The calibration curve

Figure 1. Paper area against mass, with the fitted regression line and its 95% confidence band.

Residual checks

Figure 2. Base R diagnostic plots for the fitted model — residuals vs. fitted, normal Q-Q, scale-location, and residuals vs. leverage.

Panel 1 of Figure 2 shows a funnel rather than a flat cloud: the residuals get larger as the squares get larger. Table 3 puts numbers on it.

Table 3. The largest miss at each square size. The equal-variance assumption does not hold: a balance has percentage error, not fixed error, so bigger squares miss by more.

Known area (cm²)	Largest residual (cm²)
1	0.35
4	0.59
16	0.65
36	0.96
64	1.30
100	3.09
225	2.00
400	7.28
500	7.39
567	8.76

The equal-variance assumption is therefore violated at the large-square end of the range, and confidence intervals there should be treated with caution. The leaf tracings we apply the curve to below weigh 0.1–0.4 g, at the tight end of the line, where a typical miss is a fraction of a cm².

Predicting leaf area

Table 4. Predicted leaf area for two example paper tracings.

Tracing	Mass (g)	Predicted area (cm²)
sunny-side	0.092	12.3
shady-side	0.138	18.3

The same predict() call converts a whole column at once. Table 5 applies the calibration to every leaf in our class data set, then summarizes the predicted areas by side of the tree.

Table 5. Predicted leaf area by side of the tree. Leaves whose tracings were never weighed are excluded.

Side	n	Mean area (cm²)	SD (cm²)
shady	21	36.7	5.8
sunny	19	36.0	11.5

Table 5 reports the predicted areas in the units we wanted. The two sides differ by about a square centimetre — the same story the Welch’s t-test told us about leaf mass.

Discussion

Write one short paragraph here. Did the calibration fit well — what do R² in Table 1 and Figure 1 say? What did the residual checks in Figure 2 and Table 3 show, and what does that mean for how far you would trust the curve? Why is working in area (cm²) more useful to a reader than raw tracing mass (g) when comparing sunny and shady leaves? This is where you paste the results paragraph you wrote in the activity.

References

• Whitlock, M. & Schluter, D. The Analysis of Biological Data (2nd ed.), Ch. 17.
• Wickham, H. R for Data Science (2nd ed.) — https://r4ds.hadley.nz/