kb/data/en.wikipedia.org/wiki/Blocking_(statistics)-2.md

---
title: "Blocking (statistics)"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/Blocking_(statistics)"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:49:20.919707+00:00"
instance: "kb-cron"
---

Yij is any observation for which X1 = i and X2 = j
X1 is the primary factor
X2 is the blocking factor
μ is the general location parameter (i.e., the mean)
Ti is the effect for being in treatment i (of factor X1)
Bj is the effect for being in block j (of factor X2)

=== Estimates ===
Estimate for μ : 
  
    
      
        
          
            Y
            ¯
          
        
      
    
    {\displaystyle {\overline {Y}}}
  
 = the average of all the data
Estimate for Ti : 
  
    
      
        
          
            
              Y
              ¯
            
          
          
            i
            ⋅
          
        
        −
        
          
            Y
            ¯
          
        
      
    
    {\displaystyle {\overline {Y}}_{i\cdot }-{\overline {Y}}}
  
 with 
  
    
      
        
          
            
              Y
              ¯
            
          
          
            i
            ⋅
          
        
      
    
    {\displaystyle {\overline {Y}}_{i\cdot }}
  
 = average of all Y for which X1 = i.
Estimate for Bj : 
  
    
      
        
          
            
              Y
              ¯
            
          
          
            ⋅
            j
          
        
        −
        
          
            Y
            ¯
          
        
      
    
    {\displaystyle {\overline {Y}}_{\cdot j}-{\overline {Y}}}
  
 with 
  
    
      
        
          
            
              Y
              ¯
            
          
          
            ⋅
            j
          
        
      
    
    {\displaystyle {\overline {Y}}_{\cdot j}}
  
 = average of all Y for which X2 = j.

=== Generalizations ===
Generalized randomized block designs (GRBD) allow tests of block–treatment interaction, and has exactly one blocking factor like the RCBD.
Latin squares (and other row–column designs) have two blocking factors that are believed to have no interaction.
Latin hypercube sampling
Graeco-Latin squares
Hyper-Graeco-Latin square designs

== See also ==

Algebraic statistics
Block design
Combinatorial design
Generalized randomized block design
Glossary of experimental design
Optimal design
Paired difference test
Dependent and independent variables
Blockmodeling
Paired data
Block bootstrap
Controlling for a variable

== References ==

 This article incorporates public domain material from the National Institute of Standards and Technology

== Bibliography ==
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Addelman, S. (1970). "Variability of Treatments and Experimental Units in the Design and Analysis of Experiments". Journal of the American Statistical Association. 65 (331): 1095–1108. doi:10.2307/2284277. JSTOR 2284277.
Anscombe, F.J. (1948). "The Validity of Comparative Experiments". Journal of the Royal Statistical Society. A (General). 111 (3): 181–211. doi:10.2307/2984159. JSTOR 2984159. MR 0030181.
Bailey, R. A (2008). Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. Archived from the original on 2011-03-06. Retrieved 2010-02-22.{{cite book}}:  CS1 maint: bot: original URL status unknown (link) Pre-publication chapters are available on-line.
Bapat, R. B. (2000). Linear Algebra and Linear Models (Second ed.). Springer. ISBN 978-0-387-98871-9.
Caliński T.; Kageyama S. (2000). Block designs: A Randomization approach. Vol. I: Analysis. New York: Springer-Verlag. ISBN 0-387-98578-6.
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Gates, C.E. (Nov 1995). "What Really Is Experimental Error in Block Designs?". The American Statistician. 49 (4): 362–363. doi:10.2307/2684574. JSTOR 2684574.
Kempthorne, Oscar (1979). The Design and Analysis of Experiments (Corrected reprint of (1952) Wiley ed.). Robert E. Krieger. ISBN 0-88275-105-0.
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Hinkelmann, Klaus; Kempthorne, Oscar (2008). Design and Analysis of Experiments. Vol. I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9.
Hinkelmann, Klaus; Kempthorne, Oscar (2005). Design and Analysis of Experiments. Vol. 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5.
Lentner, Marvin; Thomas Bishop (1993). "The Generalized RCB Design (Chapter 6.13)". Experimental design and analysis (Second ed.). Blacksburg, VA: Valley Book Company. pp. 225–226. ISBN 0-9616255-2-X.
Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover. ISBN 0-486-65685-3.
Raghavarao, Damaraju; Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific. ISBN 981-256-360-1.
Shah, Kirti R.; Sinha, Bikas K. (1989). Theory of Optimal Designs. Springer-Verlag. ISBN 0-387-96991-8.
Street, Anne Penfold; Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. ISBN 0-19-853256-3.
Wilk, M. B. (1955). "The Randomization Analysis of a Generalized Randomized Block Design". Biometrika. 42 (1–2): 70–79. doi:10.2307/2333423. JSTOR 2333423.
Zyskind, George (1963). "Some Consequences of randomization in a Generalization of the Balanced Incomplete Block Design". The Annals of Mathematical Statistics. 34 (4): 1569–1581. doi:10.1214/aoms/1177703889. JSTOR 2238364.