Skip to main content
Log in

Synthesis of variance

  • Published:
Psychometrika Aims and scope Submit manuscript


The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student “t” and the Fisher “z” tests to a wider range of problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  • Fisher, R. A. The fiducial argument in statistical inference.Annals of Eugenics, 1936,6, 391.

    Google Scholar 

  • Welch, B. L. The significance of the difference between two means when the population variances are unequal.Biometrika, 1938,29, 350.

    Google Scholar 

  • Hsu, P. L. Contribution to the theory of “Student's”t-test as applied to the problem of two samples.Statistical Research Memoirs, 1938,2, 1.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Satterthwaite, F.E. Synthesis of variance. Psychometrika 6, 309–316 (1941).

Download citation

  • Issue Date:

  • DOI: