Nonlinear Relationships
Students in a physics class are studying freefall to determine the relationship between the distance an object has fallen and the amount of time since release.
Time 
Distance 
Time 
Distance 
.16 
12.1 
.57 
150.2 
.24 
29.8 
.61 
182.2 
.25 
32.7 
.61 
189.4 
.30 
42.8 
.68 
220.4 
.30 
44.2 
.72 
250.4 
.32 
55.8 
.72 
261.0 
.36 
63.5 
.83 
334.5 
.36 
65.1 
.88 
375.5 
.50 
124.6 
.89 
399.1 
.50 
129.7 


1. Draw a scatter plot of the data.
2. Even though it doesn’t look linear find the regression line.
3. Find the medianmedian line.
4. Look at the residuals for both equations and find the standard error.
In a residual plot, positive and negative values should occur randomly.
The relationship between y and x need not be linear. We can examine the shape of the relationship with a scatter plot and look for more detailed information by plotting the residuals from the medianmedian line. If either the original or residual plot shows a bend and if the yversusx plot shows a generally consistent trend either up or down rather than a cup shape, we may be able to straighten the yversusx relationship by reexpressing one or both variables.
From the summary points of the medianmedian line, we compute the halfslopes:
_{}
and then find the halfslope ratio: _{}. If the halfslopes are equal, then the relationship is linear and the halfslope ratio is 1. If the halfslope ratio is not close to one, then reexpressing x or y or both may help. If the halfslope ratio is negative, the halfslopes have different signs, and reexpression will not help.
If the halfslopes are not equal, the two line segments will meet and form an angle. We can think of the angle as an arrowhead that points toward reexpressions on the ladder of powers that might make the relationship straighter. To determine how we might reexpress y, we need to know if the arrow points more upward—towards higher values of y—or more downward—toward lower yvalues. To determine how we might reexpress x, we need to know if the arrow points more to the right—toward higher x values—or more to the left—toward lower x values.
The rule for selecting a reexpression to straighten a plot is that we consider moving the expression of y or x in the direction the arrow points. That is if the arrow points down toward lower y we might try reexpressions of y lower in the ladder of powers. If the arrow points to the right, toward higher x, we might try reexpressions of x higher on the ladder of powers.
The halfslopes will suggest reexpressions for both x and y. We may choose to reexpress either y or x or both.
1. Do this for our data.
Ladder of Powers
p 
Reexpression 
Name 
Notes 

_{} 

Higher powers can be used. 
3 
_{} 
Cube 

2 
_{} 
Square 
One of the most commonly used powers 
1 
_{} 
“Raw” 
No reexpression necessary 
½ 
_{} 
Square root 
A commonly used power, especially for counts 
(0) 
_{} 
Logarithm 

–½ 
_{} 
Reciprocal root 
The minus sign preserves order. 
1 
_{} 
Reciprocal 

2 
_{} 
Reciprocal square 


_{} 

Lower powers can be used. 
1. Let’s reexpress our data.
2. Re try the fit.
3. There are other ways to think of reexpression.
Gas Mileage and Displacement for Some 1976 Automobiles
Automobile 
mpg 
Displacement 
Mazda RX4 
21.0 
160.0 
Mazda RX4 Wagon 
21.0 
160.0 
Datsun 710 
22.8 
108.0 
Hornet 4Drive 
21.4 
258.0 
Hornet Sportabout 
18.7 
360 
Valiant 
18.1 
225 
Plymouth Duster 
14.3 
360 
Mercedes 240D 
24.4 
146.7 
Mercedes 230 
22.8 
140.8 
Mercedes 280 
19.2 
167.6 
Mercedes 280C 
17.8 
167.6 
Mercedes 450SE 
16.4 
275.8 
Mercedes 450SL 
17.3 
275.8 
Mercedes 450SLC 
15.2 
275.8 
Cadillac Fleetwood 
10.4 
472 
Lincoln Continental 
10.4 
460 
Chrysler Imperial 
14.7 
440 
Fiat 128 
32.4 
78.7 
Honda Civic 
30.4 
75.7 
Toyota Corolla 
33.9 
71.1 
Toyota Corona 
21.5 
120.1 
Dodge Challenger 
15.5 
318.0 
AMC Javelin 
15.2 
304 
Camaro Z28 
13.3 
350 
Pontiac Firebird 
19.2 
400 
Fiat X19 
27.3 
79 
Porsche 9142 
26.0 
120.3 
Lotus Europa 
30.4 
95.1 
Ford Pantera L 
15.8 
351 
Ferrari Dina 1973 
19.7 
145 
Maserati Bora 
15.0 
301 
Volvo 142E 
21.4 
121 