Hartley's test

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In statistics, Hartley's test, also known as the Fmax test or Hartley's Fmax, is used in the analysis of variance to verify that different groups have a similar variance, an assumption needed for other statistical tests. It was developed by H. O. Hartley, who published it in 1950.[1]

The test involves computing the ratio of the largest group variance, max(sj2) to the smallest group variance, min(sj2). The resulting ratio, Fmax, is then compared to a critical value from a table of the sampling distribution of Fmax.[2][3] If the computed ratio is less than the critical value, the groups are assumed to have similar or equal variances.

Hartley's test assumes that data for each group are normally distributed, and that each group has an equal number of members. This test, although convenient, is quite sensitive to violations of the normality assumption.[4] Alternatives to Hartley's test that are robust to violations of normality are O'Brien's procedure,[4] and the Brown–Forsythe test.[5]

Related tests[edit]

Hartley's test is related to Cochran's C test[6][7] in which the test statistic is the ratio of max(sj2) to the sum of all the group variances. Other tests related to these, have test statistics in which the within-group variances are replaced by the within-group range.[8][9] Hartley's test and these similar tests, which are easy to perform but are sensitive to departures from normality, have been grouped together as quick tests for equal variances and, as such, are given a commentary by Hand & Nagaraja (2003).[10]

See also[edit]


  1. ^ Hartley (1950)
  2. ^ David (1952)
  3. ^ Pearson & Hartley (1970), Table 31
  4. ^ a b O'Brien (1981)
  5. ^ Keppel & Wickens (2004)
  6. ^ Cochran (1941)
  7. ^ Pearson & Hartley (1970), page 67 and Table 31a
  8. ^ Bliss et al. (1956)
  9. ^ Pearson & Hartley (1970), page 58–9 and Tables 31b,c
  10. ^ Hand & Nagaraja (2003) Section 9.7


  • Bliss, C.I., Cochran, W.G., Tukey, T.W. (1956) A Rejection Criterion Based upon the Range. Biometrika, 43, 418–422.
  • Cochran, W.G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Annals of Eugenics, 11, 47–52
  • Hand, H.A. & Nagaraja, H.N. (2003) Order Statistics, 3rd Edition. Wiley. ISBN 0-471-38926-9
  • Hartley, H.O. (1950). The Use of Range in Analysis of VarianceBiometrika, 37, 271–280.
  • David, H.A. (1952). "Upper 5 and 1% points of maximum F-ratio." Biometrika, 39, 422–424.
  • O'Brien, R.G. (1981). A simple test for variance effects in experimental designs. Psychological Bulletin, 89, 570–574.
  • Keppel, G. and Wickens, T.D. (2004). Design and analysis (4th ed.). Englewood Cliffs, NJ: Prentice-Hall.
  • Pearson, E.S., Hartley, H.O. (1970). Biometrika Tables for Statisticians, Vol 1, CUP ISBN 0-521-05920-8

External links[edit]

  • Table of critical values for the Fmax test [1]