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Algebraically speaking -

sw2 =    1 {Σai2 - 2Σaimi + 2Σmi2)
2(n - 1)


  • sw is the weighted standard deviation,
  • is the average cluster size (Σmi/n),
  • mi is the number of secondary units in each cluster,
  • n is the number of clusters in the sample,
  • is the overall proportion,
  • ai is the number of individuals with the characteristic of interest in cluster i.