Square and reciprocal data transformations

Which values should be transformed before the data are checked for linearity?

Square and reciprocal coordinates

A transformed plot uses the new explanatory values as horizontal coordinates. Equal raw x steps need not produce equal transformed steps

1. 1Keep the response unchanged For these General transformations, replace the explanatory coordinate x by t=x² or t=1/x. Continue to plot the original response y.
2. 2Use all the points Shape suggests a candidate; it does not prove linearity. For distinct transformed inputs, compare every consecutive slope. Exact points are collinear only when all these slopes agree \(\displaystyle m_i=(y_{i+1}-y_i)/(t_{i+1}-t_i)\)
3. 3State domains The reciprocal requires x≠0. Positive-input models use x>0. Squaring maps x and −x to the same coordinate; check the context before recovering an original input
4. 4Distinguish exact from measured data Equal slopes test the supplied exact observations. Measured data may only be approximately linear; an apparent trend alone cannot establish a rule for every input

Original input and transformation

\(\displaystyle x=2,\quad t=x^2\)

Apply the square transformation to the original input shown

Transformed coordinates

\(\displaystyle (t,y)=(1,2),(4,5)\)

Calculate the slope between these two transformed points

Original observations

\(\displaystyle (x,y)=(1,2),(2,8),(3,18)\)

Which row applies the square transformation to the explanatory values while keeping the response values unchanged?

Data or rule

\(\displaystyle (t,y)=(1,3),(4,9),(9,19)\)

Find the slope between the last two transformed points shown

All four original observations

\(\displaystyle (x,y)=(1,3),(2,9),(3,19),(4,34)\)

After adding the fourth observation, does plotting y against x² still give an exactly linear relationship?

Data or rule

\(\displaystyle t=1/x,\quad x=4\)

Transform the original input x=4 using the reciprocal

Data or rule

\(\displaystyle (t,y)=(1/2,6),(1/4,3)\)

Find the slope between these reciprocal transformed points

Which input is excluded from a reciprocal transformation?

Compare original inputs 2, 3 and 6. Read t and y, and observe the point on the transformed plot. Displayed decimals are rounded; use exact fractions for slope calculations

When the original positive input increases, why does P move left?

The square transform maps x=−3 and x=3 to the same t=9. What follows?

Produce and compare the transformed plots

Use the exact observations: the response is 18 at input 1, 9 at input 2 and 6 at input 3. Make one table using squared inputs and another using reciprocal inputs, keeping each response unchanged. Draw both scatterplots with labelled axes and useful windows. Compare consecutive slopes, select the linearising candidate, explain why the decreasing raw trend alone is insufficient, and distinguish the origin on the reciprocal line from an observed input

Model response: Produce and compare the transformed plots

For original inputs 1, 2 and 3 with respective responses 18, 9 and 6, compare your square and reciprocal tables and plots, slope checks and conclusion with this model

1. 1Square candidate The transformed pairs are (1,18),(4,9),(9,6). Slopes are −3 and −3/5, so they are not collinear. A window 0≤t≤10, 0≤y≤20 contains all points; label t=x² horizontally
2. 2Reciprocal candidate The transformed pairs are (1,18),(1/2,9),(1/3,6). Both consecutive slopes are 18. A window 0≤t≤1.1, 0≤y≤20 shows the line y=18t; label t=1/x horizontally \(\displaystyle (-9)/(-1/2)=18,\quad(-3)/(-1/6)=18\)
3. 3Conclude with limits The reciprocal candidate fits every supplied observation; a decreasing trend on its own does not establish that. All original inputs are positive. The transformed origin t=0 is approached as x grows, but no finite positive x maps to it. Three observations do not prove a practical law for all x.