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Chi-Square Test

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Definition of Chi-Square Test

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A statistical hypothesis test that measures how well observed data fits with the expected distribution of the data, assuming the null hypothesis is true.

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When to use Chi-Square Test for Goodness of Fit

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It is used to determine whether sample data is consistent with a population with a specific distribution.

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When to use Chi-Square Test for Independence

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It is used to determine whether there is a significant association between two categorical variables.

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Formula for Chi-Square Statistic

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The formula is

χ2=(OiEi)2Ei\chi^2 = \sum{\frac{(O_i - E_i)^2}{E_i}}
where OiO_i is the observed frequency and EiE_i is the expected frequency.

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Degrees of Freedom in Chi-Square Test

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Degrees of freedom for the Chi-Square test is the number of categories minus one for goodness of fit, or (rows - 1) * (columns - 1) for a test of independence.

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Expected Frequency Calculation

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Expected frequencies are calculated based on the null hypothesis and the distribution of the sample data.

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Null Hypothesis in Chi-Square Tests

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In chi-square tests, the null hypothesis asserts that there is no difference between the observed and expected frequencies, or no association between variables.

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Chi-Square Test P-value

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The p-value assesses the evidence against the null hypothesis, with values less than the significance level indicating a rejection of the null hypothesis.

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Limitations of Chi-Square Test

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It requires a large enough sample size for accuracy, cannot be used for small expected frequencies, and is not suitable for ordinal or continuous data without categorization.

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Assumptions of the Chi-Square Test

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Data must be randomly sampled, observations should be independent of each other, and expected frequencies should be at least 5.

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Chi-Square Distribution

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The distribution of the chi-square statistic under the null hypothesis, which is dependent on degrees of freedom.

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Contingency Table

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A data table that displays frequencies for two categorical variables and is used in chi-square tests of independence.

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Post-hoc Analyses in Chi-Square Tests

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Used to explore which specific categories or cells contribute to the significance in the chi-square test, in the case of a significant overall result.

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Effect Size in Chi-Square Tests

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Measures the strength of association or the practical significance of findings, not just the probability of results occurring by chance.

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Reporting Chi-Square Test Results

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Results are reported with the chi-square statistic value, degrees of freedom, p-value, and possibly an effect size measure, alongside a conclusion regarding the null hypothesis.

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