MDM4U1
Unit 6 Day 3
2-Variable Statistics
Non-Linear Regression
For relationships that are not linear, non-linear regression is an analytical technique for
finding the curve of best fit. Calculations for curves are much more complicated than for
straight lines.
The graphing calculator has built in regression functions for a variety of curves. They can be
used to calculate the coefficient of determination, r2, which is used to determine how well a
curve fits the data.
2
The definition for r is:
variation in y explained by variation in x
r =

total variation in y
2
 y  y 
 y  y 
2
est
2
where y is the mean y value of the data, yest is the value estimated by the best-fit curve for a
given value of x, and y is the actual observed value for a given value of x.
The coefficient of determination can have values from 0 to 1. If the curve is a perfect fit, yest
and y will be identical for each value of x and r2 will equal 1.
Conversely, if it is a poor fit,  yest  y  will be much smaller than  y  y  and r2 will be
close to zero. For any type of regression, the curve of best fit will be the one that has the
highest coefficient of determination.
2
.
2
MDM4U1
Unit 6 Day 3
2-Variable Statistics
Non-Linear Regression
TYPES OF NON-LINEAR REGRESSION
EXPONENTIAL REGRESSION:
These have equations of the form y  ab x or y  ae kx where e is an irrational number
approximately equal to 2.718…
POWER AND POLYNOMIAL REGRESSION:
In power regressions, the curve of best fit has the form y  ax b . An example of a polynomial
regression is a quadratic regression with equation in the form y=ax2 + bx + c.
Example 1
A laboratory technician monitors the growth of a bacterial culture by scanning it every hour
and estimating the number of bacteria. The initial population is unknown.
0 1 2 3 4
5
6
7
Time (h)
Population ? 10 21 43 82 168 320 475
a) Create a scatter plot on Google Sheets (or Excel or Fathom) and classify the linear
correlation.
b) Determine the correlation coefficient and line of best fit.
c) Does the line of best fit seem satisfactory?
d) Do a polynomial regression to find a curve of best fit. Also state the coefficient of
determination.
e) Does the polynomial regression seem satisfactory?
f) Repeat for an exponential regression to find a curve of best fit. State the equation and
coefficient of determination.
Which regression model do you think best fits this data? Explain.