# Chi-square statistics for Cross Tabulation and Chi-Square

Find definitions and interpretation guidance for every statistic that is provided with the chi-square test.

## Chi-Square statistic

Minitab performs a Pearson chi-square test and a likelihood-ratio chi-square test. Each chi-square test can be used to determine whether or not the variables are associated (dependent).
Pearson chi-square test

The Pearson chi-square statistic (χ2) involves the squared difference between the observed and the expected frequencies.

Likelihood-ratio chi-square test

The likelihood-ratio chi-square statistic (G2) is based on the ratio of the observed to the expected frequencies.

### Interpretation

Use the chi-square statistics to test whether the variables are associated.

In these results, both chi-square statistics are very similar. Use the p-values to evaluate the significance of the chi-square statistics.

Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

When the expected counts are small, your results may be misleading. For more information, see the Data considerations for Cross Tabulation and Chi-Square.

## DF

The degrees of freedom (DF) is the number of independent pieces of information on a statistic. The degrees of freedom for a table is (number of rows – 1), multiplied by (number of columns – 1).

### Interpretation

Minitab uses the degrees of freedom to determine the p-value associated with the test statistic.

In these results, the degrees of freedom (DF) is 4.
Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

## P-value

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Use the p-value to determine whether to reject or fail to reject the null hypothesis, which states that the variables are independent.

Minitab uses the chi-square statistic to determine the p-value.

###### Note

Minitab does not display the p-value when any expected count is less than 1 because the results can be invalid.

### Interpretation

To determine whether variables are independent, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association between the variables exists when there is no actual association.
P-value ≤ α: The variables have a statistically significant association (Reject H0)
If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that there is a statistically significant association between the variables.
P-value > α: Cannot conclude that the variables are associated (Fail to reject H0)
If the p-value is larger than the significance level, you fail to reject the null hypothesis because there is not enough evidence to conclude that the variables are associated.

In these results, the p-value is 0.019. Because the p-value is less than α, you reject the null hypothesis. You can conclude that the variables are associated.

Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019
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