# Blocking (statistics)

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In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) which are similar to one another. For example, an experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male and female patients in a double blind trial. The sex of the patient is a blocking factor accounting for treatment variablility between males and females. This reduces sources of variability and thus leads to greater precision. Suppose we have invented a process which we believe makes the soles of shoes last longer, and we wish to conduct a field trial. One possible design would be to have a group of n volunteers, give 0.5n of them shoes with the new soles and .5n of them regular shoes, randomizing (see randomization) the assignment of the two types of shoes. This type of experiment is a completely randomized design. We can then let both groups use their shoes for a suitable period of time and then compare them. A better design would be to give each person one regular sole and one new sole where random assignment of the two treatments to the left or right shoe of each volunteer is conducted. Such a design is called a randomized complete block design. This design will be more sensitive than the first, because each person is acting as their own control and thus the control group is more closely matched to the treatment group. The theoretical basis of blocking is the following mathematical result. Given random variables, X and Y

$\operatorname{var}(X-Y)= \operatorname{var}(X) + \operatorname{var}(Y) - 2\operatorname{cov}(X,Y).$

The difference between the treatment and the control can thus be given minimum variance (i.e. maximum precision) by maximising the covariance (or the correlation) between X and Y.