Goodness of Fit (Chi-square) Test

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The chi-square goodness-of-fit test can be used to test the null hypothesis that the population distribution from which the data sample is drawn is the same as the hypothesized distribution.


  • The sample values are independent, and identically distributed.
  • The sample values are grouped in C categories, and the counts of the number of sample values occurring in each category is recorded.
  • The hypothesized distribution is specified in advance, so that the number of observations that should appear each category, assuming the hypothesized distribution is the correct one, can be calculated without reference to the sample values.


  • Ways to detect before performing the chi-square goodness-of-fit test whether your data violate any assumptions.
  • Ways to examine chi-square goodness-of-fit test results to detect assumption violations.
  • Possible alternatives if your data or chi-square goodness-of-fit test results indicate assumption violations.

To properly analyze and interpret results of the chi-square goodness-of-fit test, you should be familiar with the following terms and concepts:

If you are not familiar with these terms and concepts, you are advised to consult with a statistician. Failure to understand and properly apply chi-square goodness-of-fit test may result in drawing erroneous conclusions from your data. Additionally, you may want to consult the following references:

  • Agresti, A. 1990. Categorical Data Analysis. New York: John Wiley & Sons.
  • Cochran, W. G. 1954. Some methods of strengthening the common chi-square tests. Biometrics 10: 417-451.
  • Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed. New York: John Wiley & Sons.
  • D'Agostino, R. B. and Stephens, M. A., eds. 1986. Goodness-of-fit Techniques. New York: Dekker.
  • Daniel, Wayne W. 1978. Applied Nonparametric Statistics. Boston: Houghton Mifflin.
  • Daniel, Wayne W. 1995. Biostatistics. 6th ed. New York: John Wiley & Sons.
  • Koehler, K. J. and Larntz, K. 1980. An empirical investigation of goodness-of-fit statistics for sparse multinomials. Journal of the American Statistical Association 75: 336-344.
  • Roscoe, J. T. and Byars, J. A. 1971. An investigation of the restraints with respect to sample size commonly imposed on the use of the chi-square statistic. Journal of the American Statistical Association 66: 755-759.
  • Rosner, Bernard. 1995. Fundamentals of Biostatistics. 4th ed. Belmont, California: Duxbury Press.
  • Sokal, Robert R. and Rohlf, F. James. 1995. Biometry. 3rd. ed. New York: W. H. Freeman and Co.
  • Zar, Jerrold H. 1996. Biostatistical Analysis. 3rd ed. Upper Saddle River, NJ: Prentice-Hall.

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