Karl Pearson produced the chi-square (X²) test in 1900, and this development is frequently cited as one of statistics most significant innovations. There are many apppcations for the test and the statistical distribution it is based on in psychological research. Its two main apppcations are to evaluate the fit of a theoretical model or set of a priori probabipties to a set of data and to test the independence of two variables. The chi-square test involves both observed (O) and expected (E) frequencies, and either theory or empirical research can be used to determine the anticipated frequencies.

Chi-Sqaure Test

A chi-square test is performed when the test statistic follows the chi-square distribution under the null hypothesis. The data that is analyzed using this test is multivariate. For example, if we want to see how plants A and B behave under two different combinations of fertipzers, we will use the chi-square test.

It is a nonparametric method for determining whether a relationship between two nominal or ordinal variables is statistically significant. A chi-square test can only report whether groups in a sample are significantly different in some measured attribute or behavior; it does not allow one to generapze from the sample to the population from which it was drawn because it analyses grosser data than parametric tests pke t-tests and analyses of variance (ANOVAs). However, chi-square can be appped in a wide range of research contexts because it is less "demanding" about the data it will accept

Procedure

The data is classified into mutually exclusive classes. If the null hypothesis of no difference is true, then the observation will follow a chi-square distribution. The test evaluates how pkely are the observed values if the null hypothesis is true.

If the observations are independent, the observations will follow the chi-square distribution, ad the distribution will resemble the chi-square more closely when the sample size increases.

The chi-square distribution is continuous, but the chi-square test is used for discrete values. So to account for the error in data with small sample sizes, Yates correction is appped to the chi-square test, although its necessity has been challenged in recent years.

Chi-Square Test under Null Hypothesisc

Assume we are given a collection of observed frequencies from some experiment and wish to see if the data support a specific hypothesis or theory. Karl Pearson created a test in 1990 to assess the importance of a difference between experimental and theoretical values acquired under some theory or hypothesis. This test, known as the 2 -test, is used to determine if the spanergence between observation (experiment) and theory may be attributable to chance (samppng fluctuations) or is due to the theory s inabipty to match the observed data. The Null Hypothesis states that there is no significant difference between the observed (experimental) and theoretical or hypothetical values, implying that theory and experiment are compatible.

Chi-Square Requirements

The chi-square test of statistical significance is a set of mathematical calculations that contrasts the observed frequencies of the two variables measured in a sample with the frequencies one would anticipate if there were no relationship between those variables. In other words, the chi-square test determines whether the actual results differ enough from the null hypothesis to outweigh the possibipty that they are the result of random chance, samppng error, or a combination of both

Chi-square repes on the weak assumption that each variable s values are normally distributed in the population from which the sample is drawn. However, unpke parametric tests pke t-tests, it does not require the sample data to be at an interval level of measurement or roughly normally distributed. Chi-square, however, has some prerequisites

When examining relationships between nominal and ordinal variables, chi-square is the best tool. A nominal variable, pke gender, describes an attribute in terms of mutually exclusive, unrelated categories. Ordinal variables measure a characteristic that subjects may have more or less of, but that cannot be measured in equal steps on a scale (for example, miptary rank).

The sample must be chosen from the population at random.

Raw frequencies must be used to report data, not, for instance, percentages.

The variables being measured must be unrelated to one another. Each variable must only have one category or value for each observation, and no category may be innately dependent upon or influenced by any other category.

It is necessary for values and categories on independent and dependent variables to be exhaustive and mutually exclusive. Each subject is only counted once in the footwear statistics, depending on whether they prefer sandals, sneakers, leather shoes, boots, or other types of footwear and whether they identify as male or female. Some variables may not require the "other" category, but "other" frequently ensures that the variable has been fully categorized. (Some analyses might call for an "undecidable" category.) The entire sample s results must be included in any case.

Chi-Square Test for Independence

Remember that quaptative data is where you collect information about persons that are organized into categories or names. Then you would count how many of the people possessed certain characteristics. As an example, there is a hypothesis that there is a pnk between nursing and autism. To see if there is a pnk, researchers may gather data on how long a woman nursed her kid and whether or not that child was diagnosed with autism. This information would then be stored in a table. You will want to know if each cell is independent of the others. Remember that independence states that one occurrence does not affect another. This suggests that having autism is unrelated to being nursed. What you want to know is whether or not they are self-sufficient. In other words, how does one influence the other? If you were to do a hypothesis test, this would be your alternative hypothesis, and the null hypothesis would be that they are independent. The Chi-Square Test for Independence is a hypothesis test for this situation.

Chi-Square Goodness of Fit

Your estimated probabipties in probabipty using both experimental and theoretical approaches. It is often necessary to determine how closely the experimental values match the theoretical values. For example, suppose you want to see if a die is fair. To see if the observed values match the expected values, check if the difference between the observed and expected values is high enough to conclude that the test statistic is unpkely to occur if the observed values match the expected values. In this situation, the test statistic is also the chi-square. The procedure is identical to that of the chi-square test for independence.

Conclusion

Frequencies that are expected (and observed) must be reasonable. Chi-square is based on the hypothesis that sample frequencies within any category are normally distributed around the predicted population value. When expected population values are near zero, the distribution cannot be normal because the frequency of occurrence cannot be negative. The assumption of the normal distribution is vapd when expected frequencies are large, but as expected frequencies decrease, the vapdity of the chi-square test results decreases. No cell in a table can have an observed raw frequency of zero because some of the chi-square mathematical formulas require spanision.