Variation refers to the difference between observed and expected counts. Chi-square tests assess whether this variation is due to random chance or indicates a significant relationship.

As sample size increases, standard deviation decreases. This inverse relationship is crucial for statistical inference.

As sample size increases, the sample mean gets closer to the true population mean.

A low p-value (e.g., < 0.05) suggests that the observed difference is unlikely due to chance, leading to the rejection of the null hypothesis.

A high p-value (e.g., > 0.05) suggests that the observed difference could easily be due to chance, leading to a failure to reject the null hypothesis.

Larger sample sizes increase the power of a test because they lead to smaller standard deviations, making it easier to detect a true effect if it exists.

Statistical Significance: A result is statistically significant if the p-value is below the significance level. | Practical Significance: Considers whether the magnitude of the effect is meaningful in the real world, regardless of statistical significance.

Goodness-of-Fit: Tests if the distribution of a categorical variable matches a claimed distribution. | Test for Independence: Tests if there is an association between two categorical variables.

Null Hypothesis: A statement of no effect or no difference. | Alternative Hypothesis: A statement that contradicts the null hypothesis, suggesting there is a significant effect or difference.

Type I error: Rejecting the null hypothesis when it is true (false positive). | Type II error: Failing to reject the null hypothesis when it is false (false negative).

<math-inline>\chi^2 = \sum \frac{(O - E)^2}{E}, where O is observed frequency and E is expected frequency.

Expected Count = (Row Total * Column Total) / Grand Total

df = Number of categories - 1

df = (Number of rows - 1) * (Number of columns - 1)