IIn the present communication, the application of the 𝝌²test is explored for small samples and the results are compared to the t-test and the Takiar Z test. The validity of the applicable Yates correction is also explored. The independent pairs of samples of size 12, 15, 18, 20, 24 and 30 are drawn from the Normal, Uniform and Exponential Population and compared for the possible significant differences or distribution differences. Thus, for each sample size, 500 mean comparisons are done by the t-test and the Takiar Z test and the distribution comparison are done by the Chi-square test. Proceeding, in a similar way, for the Uniform and Exponential Populations, the sample comparisons are made. Overall, the study pertains to 9000 samples comparisons.
The results surprisingly shows that the application of simple 𝝌² test picks up correctly about 73.5% of the significant differences in the normal samples as against 64.3% picked up by the 𝝌² test with the Yates correction. This clearly shows that the application of Yates correction is not beneficial. This is also observed to be true in the case of sample comparisons taken from the Uniform and Exponential distribution. The comparison of the performance of the t-test and the Takiar Z test for small samples reveals that the Takiar Z test out-perform at each sample size and in case of each sample comparisons. Relatively, the validity is better in the case of Takiar Z test to the tune of 6-8%. Further, the performance of the Takiar Z test is found to be superior to 𝝌² test and the t-test irrespective of the sample size and the type of population. Thus, for small samples Takiar Z test is recommended.
Key Words: 𝝌² test, Small samples, Takiar Z test, t-test, Normal samples, Uniform samples, Exponential samples.