Stats Tutorial - Linear Portions of Curves for Equation:

Introduction

Basics of Excel™

Basic Statistics

Linear Regression:

Introduction

Correlation

Linear Portions

Regression Equation

Regression Errors

Using the Calibration

Limits of Detection

Outliers in Regression

Data Evaluation & Comparison


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In most cases, a scientific instrument will only behave linearly within a certain region, and outside this region, will be nonlinear. It is important to choose the correct region for linear analysis to minimise any error due to nonlinearity.

A quick and convenient way to accomplish this is to visually inspect the graph and select the data points that appear linear, as well as a few other points on either side of the linear portion, that may or may not be linear. For instance, consider the fluorescense data from the previous page, only assume data was collected to a concentration of 20 pg/mL.

Fluorescence
Intensities
Concentration
(pg/ml)
2.1 0
5.0 2
9.0 4
12.6 6
17.3 8
21.0 10
24.7 12
28.4 14
31.0 16
32.9 18
33.9 20

As you can see, the data appears linear up to a point, then at around 16 pg/mL, it begins to plateau. So as to minimize the error associated with nonlinearity, we must be careful to only include data points within the linear region. However, to maximize our sample size, we want to include as many points as possible.


To determine the linear portion, we calculate R for various of combinations data points, and observe where the point for which there is a drastic decrease in the R-value. The following table shows R-values for different values of n.

n R
7 0.9989
8 0.9992
9 0.9988
10 0.9969
11 0.9924

As you can see, the Correlation Coefficient drops off after the 9th data point, so using a 9-point best-fit line is probably the most suitable. You can determine this line, then calculate the residuals for each data point, to see if this choice is valid. This is left to the student to try. A more rigorous method to test the validity of your choice is to use a t-test, which is discussed in a later section.

IMPORTANT!! When using this method, make sure that you include ALL the data points in your plot, not only the linear ones, and then indicate on your graph that you used only the linear data for the calibration curve.


In the next few pages, we begin to explore in-depth the calculation of the linear regression equation for computing calibration curves. We also explore the error associated with the regression equation.

© 2006 Dr. David C. Stone & Jon Ellis, Chemistry, University of Toronto
Last updated: September 26th, 2006