Fisher's inequality

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In combinatorial mathematics, Fisher's inequality, named after Ronald Fisher, is a necessary condition for the existence of a balanced incomplete block design satisfying certain prescribed conditions.

Fisher, a population geneticist and statistician, was concerned with the design of experiments 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.

It was required that:

  • k different varieties are in each block, 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 \ge v.\,

[edit] References

  • R. A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, volume 10, 1940, pages 52–75.
  • R. C. Bose, "A Note on Fisher's Inequality for Balanced Incomplete Block Designs", Annals of Mathematical Statistics, 1949, pages 619–620.