Types of Statistical Tests

Understanding different statistical tests is key in analyzing data effectively. These tests help determine relationships, differences, and associations among variables, forming the backbone of statistical inference in Honors and Preparatory Statistics. Here’s a breakdown of essential tests.

1. t-test (one-sample, two-sample, paired)

   • One-sample t-test compares the mean of a single group to a known value or population mean.
   • Two-sample t-test compares the means of two independent groups to determine if they are significantly different.
   • Paired t-test compares means from the same group at different times (e.g., before and after treatment).
   • Assumes data is normally distributed and variances are equal (for two-sample).
   • Useful for small sample sizes (n < 30) when population standard deviation is unknown.

2. ANOVA (one-way, two-way)

   • One-way ANOVA tests for differences in means across three or more independent groups.
   • Two-way ANOVA examines the effect of two independent variables on a dependent variable, including interaction effects.
   • Assumes normality and homogeneity of variances among groups.
   • Helps to identify if at least one group mean is different without conducting multiple t-tests.
   • Provides F-statistic to determine significance.

3. Chi-square test

   • Tests the association between categorical variables in a contingency table.
   • Compares observed frequencies to expected frequencies under the null hypothesis.
   • Requires a minimum expected frequency in each cell (usually at least 5).
   • Can be used for goodness-of-fit tests or tests of independence.
   • Not suitable for small sample sizes or continuous data.

4. Regression analysis (simple linear, multiple)

   • Simple linear regression models the relationship between one independent variable and one dependent variable.
   • Multiple regression involves two or more independent variables predicting a dependent variable.
   • Assesses the strength and direction of relationships using coefficients.
   • Assumes linearity, independence, homoscedasticity, and normality of residuals.
   • Useful for prediction and understanding relationships between variables.

5. Correlation analysis

   • Measures the strength and direction of the linear relationship between two continuous variables.
   • Correlation coefficient (r) ranges from -1 to 1, indicating negative, no, or positive correlation.
   • Does not imply causation; correlation does not equal causation.
   • Assumes linearity and normality of the data.
   • Commonly used to identify potential relationships before further analysis.

6. Z-test

   • Used to determine if there is a significant difference between sample and population means or between two sample means.
   • Requires large sample sizes (n ≥ 30) or known population standard deviation.
   • Assumes normal distribution of the data.
   • Can be one-sample or two-sample.
   • Provides a Z-score to assess significance.

7. F-test

   • Compares variances between two or more groups to determine if they are significantly different.
   • Commonly used in ANOVA to test the equality of variances.
   • Assumes normality and independence of observations.
   • Provides an F-statistic to assess significance.
   • Useful for validating assumptions of other statistical tests.

8. Mann-Whitney U test

   • A non-parametric test that compares differences between two independent groups.
   • Does not assume normality and is used when data is ordinal or not normally distributed.
   • Ranks all data points and compares the sum of ranks between groups.
   • Useful for small sample sizes or when data violates t-test assumptions.
   • Provides a U statistic to determine significance.

9. Wilcoxon signed-rank test

   • A non-parametric test for comparing two related samples or repeated measurements on a single sample.
   • Used when the data does not meet the assumptions of the paired t-test.
   • Ranks the absolute differences between pairs and considers the direction of differences.
   • Suitable for ordinal data or non-normally distributed interval data.
   • Provides a W statistic to assess significance.

10. Kruskal-Wallis test

    • A non-parametric alternative to one-way ANOVA for comparing three or more independent groups.
    • Does not assume normality and is used for ordinal or non-normally distributed data.
    • Ranks all data points and compares the sum of ranks across groups.
    • Useful when ANOVA assumptions are violated.
    • Provides a H statistic to determine significance.