Inferential Statistics Tools
Perform hypothesis testing and inferential statistical analysis directly in your browser. StatFusion’s inferential statistics tools help you draw conclusions about populations based on sample data, with no software installation required.
Inferential statistics allows researchers to make probability-based judgments about populations using sample data. These tools help you test hypotheses, calculate p-values, and determine statistical significance for your research questions.
What is Inferential Statistics?
Inferential statistics uses sample data to make generalizations or inferences about a population. Unlike descriptive statistics, which simply summarizes data, inferential statistics allows researchers to:
- Test hypotheses about population parameters
- Estimate population parameters with confidence intervals
- Compare groups to assess if differences are statistically significant
- Determine relationships between variables and assess their significance
The tools in this section help you perform these inferential tasks with proper statistical rigor while providing clear interpretations of your results.
Available Inferential Tests
StatFusion offers a comprehensive suite of inferential statistical tests organized by category and purpose.
Mean Comparisons
Tests for comparing means between groups or against known values.
One Sample Tests
Compare a sample mean to a known or hypothesized value:
- One-Sample t-Test - Test if a sample mean differs from a specified value
- One-Sample z-Test - Compare a sample mean to a known value when population standard deviation is known
Two Sample Tests
Compare means between two groups:
- Independent Samples t-Test - Compare means between two unrelated groups
- Paired Samples t-Test - Compare means between two related measurements
- Welch’s t-Test - Compare means when variances differ
Multiple Sample Tests
Compare means across more than two groups:
- One-Way ANOVA - Compare means across multiple independent groups
- Repeated Measures ANOVA - Compare means across multiple related measurements
- Factorial ANOVA - Test effects of multiple factors and their interactions
- ANCOVA - Compare means while controlling for covariates
Post-Hoc Tests
Follow-up tests after finding significant effects in multiple comparisons:
- Tukey’s HSD - Compare all pairs of means while controlling Type I error
- Bonferroni Test - Adjust p-values for multiple comparisons
- Dunnett’s Test - Compare each group to a control group
- Scheffé’s Test - Test all possible contrasts
Non-Parametric Tests
Distribution-free alternatives when parametric assumptions aren’t met.
One Sample Tests
Compare a sample to a hypothesized distribution or value:
- Wilcoxon Signed Rank Test (Single) - Non-parametric alternative to one-sample t-test
- Sign Test - Test median against a hypothesized value
Two Sample Tests
Compare two samples without normality assumptions:
- [Mann-Whitney U Test](/apps/statfusion/analysis/inferential/non-parametric/two-sample/mann-whitney-u-test-calculator-nonparametric.qmd} - Non-parametric alternative to independent t-test
- Wilcoxon Signed Rank Test (Paired) - Non-parametric alternative to paired t-test
- Kolmogorov-Smirnov Test - Compare two sample distributions
Multiple Sample Tests
Compare more than two samples without normality assumptions:
- Kruskal-Wallis Test - Non-parametric alternative to one-way ANOVA
- Friedman Test - Non-parametric alternative to repeated measures ANOVA
- Jonckheere-Terpstra Test - Test for ordered alternatives across groups
Non-Parametric Post-Hoc Tests
Follow-up tests for non-parametric multiple comparisons:
- Dunn’s Test - Post-hoc test after Kruskal-Wallis
- Conover-Iman Test - Alternative post-hoc for non-parametric ANOVA
- Nemenyi Test - Post-hoc test after Friedman test
Correlation & Association Tests
Tests for measuring and testing relationships between variables.
Correlation Tests
Measure and test linear and monotonic relationships:
- Pearson Correlation - Test linear relationship between continuous variables
- Spearman Correlation - Test monotonic relationship (rank correlation)
- Kendall’s Tau - Non-parametric measure of ordinal association
Categorical Association Tests
Examine relationships between categorical variables:
- Chi-Square Test - Test independence between categorical variables
- Fisher’s Exact Test - Alternative to Chi-Square for small samples
- Cochran-Mantel-Haenszel Test - Test conditional independence
Proportion Tests
Tests for comparing proportions within and between samples.
Single and Two-Sample Proportion Tests
Test proportions against a reference or between groups:
- One-Sample Proportion Test - Test a proportion against a hypothesized value
- Two-Sample Proportion Test - Compare proportions between independent samples
Variance Tests
Tests for comparing variances between samples.
Two-Sample Variance Tests
Compare variances between two samples:
- F-Test - Compare variances of two normally distributed populations
- Levene’s Test - Less sensitive to departures from normality
Multiple-Sample Variance Tests
Compare variances across multiple groups:
- Bartlett’s Test - Test for equal variances across groups (assumes normality)
- Brown-Forsythe Test - Robust test for homogeneity of variance
Goodness-of-Fit Tests
Tests for determining if data follows specified distributions.
Distribution Tests
Test if data follows theoretical distributions:
- Chi-Square Goodness of Fit - Test if categorical data follows expected frequencies
- Anderson-Darling Test - Test if data follows a specified continuous distribution
Normality Tests
Specific tests for normal distribution:
- Shapiro-Wilk Test - Powerful test for normality
- Kolmogorov-Smirnov Normality Test - Compare data to normal distribution
- Q-Q Plot Analysis - Visual assessment of normality
How to Choose the Right Inferential Test
Selecting the appropriate statistical test is crucial for valid analysis:
Decision Guide for Common Inferential Tests
- First, identify your research question and type of data
- Then, check if parametric assumptions are met for your data
- Finally, select the appropriate test based on your specific scenario
Comparing Means
One sample vs. known value:
- Parametric: One-Sample t-Test
- Non-parametric: Wilcoxon Signed Rank Test (Single)
Two independent groups:
- Parametric: Independent Samples t-Test
- Non-parametric: Mann-Whitney U Test
Two related/paired groups:
- Parametric: Paired Samples t-Test
- Non-parametric: Wilcoxon Signed Rank Test (Paired)
Three+ independent groups:
- Parametric: One-Way ANOVA
- Non-parametric: Kruskal-Wallis Test
Three+ related groups:
- Parametric: Repeated Measures ANOVA
- Non-parametric: Friedman Test
Testing Relationships
Two continuous variables:
- Parametric: Pearson Correlation
- Non-parametric: Spearman Correlation
Two categorical variables:
- Large samples: Chi-Square Test
- Small samples: Fisher’s Exact Test
Comparing proportions:
- Single proportion: One-Sample Proportion Test
- Two independent proportions: Two-Sample Proportion Test
- Paired proportions: McNemar’s Test
Comparing variances:
- Two groups (normal): F-Test
- Two groups (robust): Levene’s Test
- Multiple groups: Bartlett’s Test
When to Use Parametric vs. Non-parametric Tests
Use parametric tests when:
- Your data is normally distributed
- Sample size is large enough
- Variables are measured on interval or ratio scale
- Variances are approximately equal (for certain tests)
Use non-parametric tests when:
- Your data significantly deviates from normal distribution
- Sample size is small
- Variables are measured on ordinal or nominal scale
- Data contains outliers that cannot be removed
Understanding Inferential Statistics Concepts
A p-value represents the probability of observing results at least as extreme as those in your sample, assuming the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests that your sample results would be unlikely if the null hypothesis were true, leading to its rejection.
Common misinterpretations:
- A p-value is NOT the probability that the null hypothesis is true
- A p-value is NOT the probability that your findings occurred by chance
- A p-value does NOT measure the size or importance of an effect
Statistical power is the probability that a test will correctly reject the null hypothesis when the alternative hypothesis is true. In simpler terms, it’s the likelihood of detecting an effect when one truly exists.
Power is affected by:
- Sample size (larger samples increase power)
- Effect size (larger effects are easier to detect)
- Significance level (a less stringent level increases power)
- Variability in the data (less variability increases power)
A power of 0.8 (80%) is commonly targeted in research design, meaning you have an 80% chance of detecting an effect if it exists.
Type I Error (False Positive): Rejecting the null hypothesis when it is actually true. The probability of a Type I error is equal to your significance level (alpha), typically 0.05.
Type II Error (False Negative): Failing to reject the null hypothesis when it is actually false. The probability of a Type II error is denoted as beta, and 1-beta equals statistical power.
These errors represent a trade-off: decreasing one type of error typically increases the other. Researchers must balance both based on the relative costs of each error type in their specific context.
Effect size measures the magnitude of an observed effect, independent of sample size. Unlike p-values, which can be significant for trivial effects with large samples, effect sizes help quantify how meaningful a difference or relationship is in practical terms.
Common effect size measures include:
- Cohen’s d (for t-tests): ~0.2 (small), ~0.5 (medium), ~0.8 (large)
- Eta-squared or Partial eta-squared (for ANOVA): ~0.01 (small), ~0.06 (medium), ~0.14 (large)
- Correlation coefficient r: ~0.1 (small), ~0.3 (medium), ~0.5 (large)
Reporting effect sizes alongside p-values provides a more complete picture of your results and facilitates meta-analysis.
Statistical significance indicates that an observed effect is unlikely to have occurred by chance, based on a p-value threshold (usually p < 0.05). However, with large samples, even tiny, practically meaningless effects can be statistically significant.
Practical significance refers to whether an effect is large enough to matter in a real-world context. This judgment depends on the specific field, research question, and potential applications.
Best practice is to consider both: use statistical significance to determine if an effect is reliable, and effect size and domain knowledge to evaluate practical significance.
Common Mistakes in Inferential Statistics
Avoid these common pitfalls when conducting inferential analyses:
- p-hacking: Performing multiple tests and only reporting significant results
- HARKing (Hypothesizing After Results are Known): Presenting post-hoc hypotheses as if they were a priori
- Ignoring assumptions: Failing to check if your data meets test assumptions
- Misinterpreting p-values: Treating p > 0.05 as “proving” the null hypothesis
- Neglecting effect sizes: Focusing only on p-values without considering magnitude
- Inappropriate test selection: Using tests that don’t match your data or research question
- Overlooking multiple comparisons: Not adjusting for increased Type I error risk
StatFusion’s tools are designed to help you avoid these mistakes by guiding you through appropriate test selection, assumption checking, and result interpretation.
Further Reading and Resources
Enhance your understanding of inferential statistics with these recommended resources:
- An Introduction to Statistical Learning
- Statistics How To
- UCLA’s Statistical Methods and Data Analytics Resources
- Khan Academy: Inferential Statistics
Reuse
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date = {2025-04-10},
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