Fisher's inequality
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Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.
Let:
- v be the number of varieties of plants;
- b be the number of blocks.
To be a balanced incomplete block design it is required that:
- k different varieties are in each block, 1 ≤ k < v; no variety occurs twice in any one block;
- any two varieties occur together in exactly λ blocks;
- each variety occurs in exactly r blocks.
Fisher's inequality states simply that
- b ≥ v.