Chi-Square Test (χ² – Test)

Abstract

The χ ² is used to study the divergence of the observed values from the expected values. It is also possible to study the association of any two attributes from the contingency table. The χ ² is the simplest and most popular non-parametric test. It was first used by Karl Pearson in the year 1900. The test is used to decide whether the discrepancy between theory and experiment is significant or not, i.e., to test whether the differences between theoretical (expected) and observed values could be attributed to a chance or not.

In sampling studies, one cannot expect perfect matching of observed and expected frequencies. The difference between observed and expected frequencies may arise due to fluctuations of sampling. It is therefore necessary to determine the extent of difference that can be ignored due to chance occurrence. The values of χ ² under conditions are available in the shape of table at the end of this chapter. If the actual value of χ ² is more than that given in the table, for a significant value of “p,” then it would indicate that the difference in expected and observed values is not only due to chance fluctuations, but it could be due to some other reasons.

Formula

$$ {\boldsymbol{\chi}}^2=\sum \frac{{\left(O-E\right)}^2}{E} $$

Where O = observed value and E = expected value

The expected frequency can be calculated with the help of the following formula:

$$ \mathrm{Expected}\kern0.5em \mathrm{value}=\frac{R\times C}{N} $$

Where R = row total, C = column total, and N = total number of observations.

If the value of χ ² computed is less than that given in the table, then it would be acceptable that the difference has arisen only due to chance fluctuations at a particular level of significance. Thus, it is possible to conclude about “goodness of fit” and also about “association” as illustrated in the contingency table.