Statistical Framework 

for Comparing Lumber 

Sorting Procedures

Steve Verrill

Richard A. Johnson

James W. Evans

United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL–RP–651
July 2009

Verrill, Steve; Johnson, Richard A.; Evans, James W. 2009. Statistical 

framework for comparing lumber sorting procedures. Research Paper 

FPL-RP-651. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 17 p.

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Abstract

In this paper, we take a preliminary look at techniques for 

comparing sorts that yield grades of lumber. We propose 

methodology that takes into account differences in grade 

prices and the costs associated with misgrading. We focus 

on two- and three-category sorts, but our results could be 

readily extended. We provide web links to sample FORTRAN implementations of this methodology for the case in 

which the strength predictor and strength have a bivariate 

normal distribution and the load distribution is also normally distributed. We indicate how these approaches would 

have to be modified for other predictor, strength, and load 

distributions. 

Keywords: lumber grading, correlation estimation, comparing sorting procedures, classification 

Contents

Page

1 Introduction........................................................................ 1

2 Maximizing Expected Value .............................................. 1

3 Bivariate Normal, Two-Bin Case....................................... 2

 Probability Density Function (pdf) of the Lower

 Bin Strengths..................................................................... 2

 Probability of a Lower Bin Failure ................................... 3

 Upper Bin Strengths and Probability of Failure ............... 4

 Value.................................................................................. 5

4 Bivariate Normal, Three-Bin Case..................................... 5

5 Lognormal/Normal Case.................................................... 7

6 Non-Normal Load Distributions........................................ 7

7 Pre-Specified Load Distributions....................................... 7

8 Estimating the Correlation, ρ, between a Predictor 

and Strength .......................................................................... 8

9 Estimating the Correlation between Two 

Predictors — Aside ............................................................... 8

10 Uncertainty in the Value of a Sort.................................... 9

11 Example Computer Programs........................................ 10

12 An Alternative Model..................................................... 11

13 Summary........................................................................ 11

References........................................................................... 12

Figures................................................................................ 13
 	

	 
	 

	 
	


	 	
	

  	
 	


  	  
 	



	

      	


  	

 
 	



   	
 	


  	  
 	



 	



  
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Value

0.0 0.2 0.4 0.6 0.8 1.0

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Correlation = 0.6

Breakpoint probability (q)

Value

0.0 0.2 0.4 0.6 0.8 1.0

0.4 0.6 0.8 1.0 1.2

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Correlation = 0.8

Breakpoint probability (q)

Value

0.0 0.2 0.4 0.6 0.8 1.0

0.4 0.6 0.8 1.0 1.2

  	

   
 
 

    

    



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Correlation = 0.6

Breakpoint probability (q)

Value

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

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Correlation = 0.7

Breakpoint probability (q)

Value

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

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Correlation = 0.8

Breakpoint probability (q)

Value

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

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Correlation = 0.60

Upper breakpoint probability (q2)

Lower breakpoint probability (q1)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.7404 0.9 0.96 0.96 0.97

0.98

0.99

0.99

0.991

0.992

0.993

0.9933

0.9933

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Correlation = 0.70

Upper breakpoint probability (q2)

Lower breakpoint probability (q1)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.8724 0.98 0.98

0.99

0.994

0.995

0.9955 0.9958

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Correlation = 0.80

Upper breakpoint probability (q2)

Lower breakpoint probability (q1)

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.9660.995 0.995

0.995

0.995

0.9982 0.9972

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