Is Weight Proportional to (Length)3?

A lowess line provides a good visual summary of the shape of a relationship, but a mathematical equation is easier to use for further analysis. What type of equation might you expect to relate slug weight and length? Since a slug's weight is likely to be roughly proportional to its volume (with units cm3), we might expect

wt propn to len3


The scatterplot below allows us to examine whether the relationship between weight and (length)3 is approximately linear.

Drag the red line on the horizontal axis towards the left to apply a power transformation to the slug lengths. A scatterplot of weight against (length)3 seems roughly linear, so our hypothesis of a linear relationship between weight and (length)3 is supported by the data.

Click Show Lowess Smooth and repeat. The lowess line also becomes close to linear when weight is plotted against (length)3.

Two scales are drawn beneath the scatterplot. The bottom scale is most relevant to the exercise above — it shows powers of length and its values are regularly spaced along the axis. The top scale is a nonlinear scale showing the corresponding untransformed values of length.