Chi-Square Test

The Chi-square test is an important non-parametric test developed by Karl Pearson in 1900 based on the distribution of any variable.

  • The test is used to measure the differences between what is observed and what is expected according to an assumed hypothesis.
  • It is also called as X² test in Greek X² is called Chi.


  • This test assumes frequencies and not on the parameters like mean and standard deviation.
  • The test is assumed for testing the hypothesis, not for estimation.
  • No rigid assumptions are necessary regarding the type of population.
  • No need for parameter values and assumes relatively fewer mathematical details.

Applications of Chi-Square Test

  • Used in Goodness of fit of distributions.
  • This test enables us to see how well does the assumed theoretical distribution (such as Binomial distribution, Poisson distribution, or Normal distribution) fits the observed.
  • Test of independence of attributes. 
  • Test enables us to explain whether or not two attributes are associated. For instance, we may be interested in knowing whether a new medicine is effective in controlling fever or not.
  • Test of homogeneity in the sample distributions.
  • This test can also be used to test whether the occurrence of events follows uniformity or not, for example, e.g, the admission of patients in a government hospital in all days of the week is uniform or not can be tested with the help of chi-square test.

Conditions under Chi-square test is used

  • The data must be in the form of frequencies.
  • The frequency data must have a precise numerical value and must be organized into categories or groups.
  • Observations recorded and used are collected on a random basis.
  • All the items in the sample must be independent.
  • The overall number of items must also be reasonably large. It should normally be at least 50.

Limitations of Chi-Square Test

  • The data is from a random sample.
  • This test applied in a four-fold table, will not give a reliable result with one degree of freedom if the expected value in any cell is less than 5.
  • The test may be misleading if any expected frequency is much below 5. in that case another appropriate test should be applied.
  • Interpret this test with caution if the sample total or total of values in all the cells is less than 50.
  • This test tells the presence or absence of an association between the events but doesn’t measure the strength of association.
  • This test doesn’t indicate the cause and effect, it only tells the probability of occurrence of association by chance.
  • The test is to be applied only when the individual observations of the sample are Independent which means that the occurrence of one individual observation (event) does not affect the occurrence of any other observation (event) in it.
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