---
title: "Design of experiments"
chunk: 2/5
source: "https://en.wikipedia.org/wiki/Design_of_experiments"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:42:27.446946+00:00"
instance: "kb-cron"
---

Comparison
In some fields of study it is not possible to have independent measurements to a traceable metrology standard.  Comparisons between treatments are much more valuable and are usually preferable, and often compared against a scientific control or traditional treatment that acts as baseline.
Randomization
Random assignment is the process of assigning individuals at random to groups or to different groups in an experiment, so that each individual of the population has the same chance of becoming a participant in the study. The random assignment of individuals to groups (or conditions within a group) distinguishes a rigorous, "true" experiment from an observational study or "quasi-experiment". There is an extensive body of mathematical theory that explores the consequences of making the allocation of units to treatments by means of some random mechanism (such as tables of random numbers, or the use of randomization devices such as playing cards or dice). Assigning units to treatments at random tends to mitigate confounding, which makes effects due to factors other than the treatment to appear to result from the treatment.
The risks associated with random allocation (such as having a serious imbalance in a key characteristic between a treatment group and a control group) are calculable and hence can be managed down to an acceptable level by using enough experimental units. However, if the population is divided into several subpopulations that somehow differ, and the research requires each subpopulation to be equal in size, stratified sampling can be used. In that way, the units in each subpopulation are randomized, but not the whole sample. The results of an experiment can be generalized reliably from the experimental units to a larger statistical population of units only if the experimental units are a random sample from the larger population; the probable error of such an extrapolation depends on the sample size, among other things.
Statistical replication
Measurements are usually subject to variation and measurement uncertainty; thus they are repeated and full experiments are replicated to help identify the sources of variation, to better estimate the true effects of treatments, to further strengthen the experiment's reliability and validity, and to add to the existing knowledge of the topic. However, certain conditions must be met before the replication of the experiment is commenced: the original research question has been published in a peer-reviewed journal or widely cited, the researcher is independent of the original experiment, the researcher must first try to replicate the original findings using the original data, and the write-up should state that the study conducted is a replication study that tried to follow the original study as strictly as possible.
Blocking
Blocking is the non-random arrangement of experimental units into groups (blocks) consisting of units that are similar to one another. Blocking reduces known but irrelevant sources of variation between units and thus allows greater precision in the estimation of the source of variation under study.

Orthogonality

Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal. Because of this independence, each orthogonal treatment provides different information to the others. If there are T treatments and T − 1 orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
Multifactorial experiments
Use of multifactorial experiments instead of the one-factor-at-a-time method.  These are efficient at evaluating the effects and possible interactions of several factors (independent variables). Analysis of experiment design is built on the foundation of the analysis of variance, a collection of models that partition the observed variance into components, according to what factors the experiment must estimate or test.

== Example ==

This example of design experiments is attributed to Harold Hotelling, building on examples from Frank Yates. The experiments designed in this example involve combinatorial designs.
Weights of eight objects are measured using a pan balance and set of standard weights.  Each weighing measures the weight difference between objects in the left pan and any objects in the right pan by adding calibrated weights to the lighter pan until the balance is in equilibrium. Each measurement has a random error 
  
    
      
        ϵ
      
    
    {\displaystyle \epsilon }
  
.  The average error is zero; the standard deviations of the probability distribution of the errors is the same number σ on different weighings; errors on different weighings are independent.  Denote the true weights by

  
    
      
        
          θ
        
        =
        (
        
          θ
          
            1
          
        
        ,
        …
        ,
        
          θ
          
            8
          
        
        )
        
      
    
    {\displaystyle \mathbf {\theta } =(\theta _{1},\dots ,\theta _{8})\,}
  
.
We consider two different experiments with the same amount of measurements:

Weigh each of the eight objects individually.

  
    
      
        
          
            
              
              
                
                  left pan
                
              
              
                
                  right pan
                
              
            
            
              
                
                  1st weighing:
                
              
              
                1
                 
              
              
                
                  (empty)
                
              
            
            
              
                
                  2st weighing:
                
              
              
                2
                 
              
              
                
                  (empty)
                
              
            
            
              
                
                  3rd weighing:
                
              
              
                3
                 
              
              
                
                  (empty)
                
              
            
            
              
                .
                .
                .
              
              
                .
                .
                .
              
              
                .
                .
                .
              
            
          
        
      
    
    {\displaystyle {\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ &{\text{(empty)}}\\{\text{2st weighing:}}&2\ &{\text{(empty)}}\\{\text{3rd weighing:}}&3\ &{\text{(empty)}}\\...&...&...\end{array}}}
  

Do the eight weighings according to the following schedule:

  
    
      
        
          
            
              
              
                
                  left pan
                
              
              
                
                  right pan
                
              
            
            
              
                
                  1st weighing:
                
              
              
                1
                 
                2
                 
                3
                 
                4
                 
                5
                 
                6
                 
                7
                 
                8
              
              
                
                  (empty)
                
              
            
            
              
                
                  2nd:
                
              
              
                1
                 
                2
                 
                3
                 
                8
                 
              
              
                4
                 
                5
                 
                6
                 
                7
              
            
            
              
                
                  3rd:
                
              
              
                1
                 
                4
                 
                5
                 
                8
                 
              
              
                2
                 
                3
                 
                6
                 
                7
              
            
            
              
                
                  4th:
                
              
              
                1
                 
                6
                 
                7
                 
                8
                 
              
              
                2
                 
                3
                 
                4
                 
                5
              
            
            
              
                
                  5th:
                
              
              
                2
                 
                4
                 
                6
                 
                8
                 
              
              
                1
                 
                3
                 
                5
                 
                7
              
            
            
              
                
                  6th:
                
              
              
                2
                 
                5
                 
                7
                 
                8
                 
              
              
                1
                 
                3
                 
                4
                 
                6
              
            
            
              
                
                  7th:
                
              
              
                3
                 
                4
                 
                7
                 
                8
                 
              
              
                1
                 
                2
                 
                5
                 
                6
              
            
            
              
                
                  8th:
                
              
              
                3
                 
                5
                 
                6
                 
                8
                 
              
              
                1
                 
                2
                 
                4
                 
                7
              
            
          
        
      
    
    {\displaystyle {\begin{array}{lcc}&{\text{left pan}}&{\text{right pan}}\\\hline {\text{1st weighing:}}&1\ 2\ 3\ 4\ 5\ 6\ 7\ 8&{\text{(empty)}}\\{\text{2nd:}}&1\ 2\ 3\ 8\ &4\ 5\ 6\ 7\\{\text{3rd:}}&1\ 4\ 5\ 8\ &2\ 3\ 6\ 7\\{\text{4th:}}&1\ 6\ 7\ 8\ &2\ 3\ 4\ 5\\{\text{5th:}}&2\ 4\ 6\ 8\ &1\ 3\ 5\ 7\\{\text{6th:}}&2\ 5\ 7\ 8\ &1\ 3\ 4\ 6\\{\text{7th:}}&3\ 4\ 7\ 8\ &1\ 2\ 5\ 6\\{\text{8th:}}&3\ 5\ 6\ 8\ &1\ 2\ 4\ 7\end{array}}}
  