A complete toolkit for rigorous statistical inference — from classical t-tests and ANOVA to non-parametric alternatives. Every test computes exact p-values with detailed interpretations.
✦ Smart Test Selector — Not sure which test? Let us guide you.
What kind of data are you working with?
This determines the family of tests available to you.
What is the goal of your comparison?
Think about how many groups and whether observations are independent or paired.
What is the goal of your analysis?
How many groups are you comparing?
Compare the mean of your sample to a hypothesized value μ₀. The t-test is appropriate when σ is unknown (which is almost always). For known σ and large n, a Z-test is equivalent.
Open t-test Tool →Also consider: Z-test if population σ is known.
Compare the means of two independent groups. Welch's t-test is recommended as it does not assume equal variances, making it more robust for most situations.
Open t-test Tool →If data is clearly non-normal: Mann-Whitney U Test
When each observation in group 1 is matched to one in group 2 (same subject before/after, matched pairs). The paired t-test analyses the differences directly, reducing within-subject variability.
Open t-test Tool →ANOVA (Analysis of Variance) tests whether at least one group mean differs from the others. Use one-way for a single factor, two-way when you have two categorical factors. Post-hoc tests identify which specific pairs differ.
Open ANOVA Tool →Non-parametric alternative: Kruskal-Wallis Test
When normality cannot be assumed — use rank-based tests that make no distribution assumptions. These are also robust to outliers and suitable for ordinal data.
Mann-Whitney U (2 groups) → Kruskal-Wallis (3+ groups) →Tests whether observed frequencies in categories match expected theoretical frequencies. Suitable for testing if data follows a uniform, Poisson, or any specified distribution.
Open Chi-Square Tool →Tests whether two categorical variables are independent in a contingency table. Also quantifies the strength of association with Cramér's V.
Open Chi-Square Tool →Test whether a proportion equals a target value, or compare two proportions from independent samples. Requires n·p ≥ 5 and n·(1-p) ≥ 5 for the normal approximation to hold.
Open Z-Test Tool →Tests whether two populations have equal variances. Commonly used as a preliminary test before a pooled two-sample t-test, or to assess process consistency.
Open F-Test Tool →For 3+ groups, the Kruskal-Wallis tool includes a Levene's-style variance comparison. Alternatively, run pairwise F-tests between groups.
Open F-Test Tool →Click any card to open the tool directly.
Compare means when the population standard deviation is unknown. Includes Welch's correction for unequal variances. Computes exact p-values, confidence intervals, and effect sizes.
Analysis of Variance for 3+ group comparisons. Full ANOVA table with SS, MS, F-statistic, and p-value. Tukey HSD post-hoc tests identify which group pairs differ significantly.
Tests for categorical data: fit to a theoretical distribution, or test independence between two categorical variables in a contingency table. Reports Cramér's V effect size.
Z-tests for large samples or known population σ. Covers one-sample mean tests, one-proportion tests against a target, and two-sample proportion comparisons.
Tests equality of variances between two groups using the F-distribution. Essential for checking the equal-variance assumption before pooled t-tests, and for process capability studies.
Non-parametric alternative to the two-sample t-test. Works for ordinal data, skewed distributions, and small samples. No normality assumption required.
Non-parametric alternative to one-way ANOVA for comparing 3+ independent groups. Based on ranks; no normality required. Includes Dunn's post-hoc test for pairwise comparisons.
A summary with real-world examples to help you choose the right test for your situation.
| Situation | Example | Assumptions | Test | Link |
|---|---|---|---|---|
| 📏 Comparing Means (Continuous Data) | ||||
| 1 sample vs. a known/target value, σ unknown | A factory claims its bags weigh 500g on average. You weigh 25 bags — is the true mean actually 500g? | Population ≈ normal, or n ≥ 30 (CLT) | One-Sample t-test | → Open |
| 1 sample vs. a known/target value, σ known | IQ scores are known to have population σ = 15. Is your sample's mean IQ different from 100? | Population σ known; normal or large n | One-Sample Z-test | → Open |
| 2 independent groups, means | Do students in Class A score differently on average than students in Class B on the same exam? | Approx. normal or n ≥ 30 per group; variances may differ | Welch's t-test | → Open |
| Paired / before-after (same subjects) | Measure each patient's blood pressure before and after a new medication — did it change? | Differences ≈ normally distributed | Paired t-test | → Open |
| 3+ independent groups, means, 1 factor | Compare average crop yield across 4 different fertilizer brands. | Normal residuals, roughly equal variances across groups | One-Way ANOVA | → Open |
| 2 factors at once (+ their interaction) | Does yield depend on both fertilizer type AND watering frequency, and do they interact? | Normal residuals, balanced design (equal n per cell) | Two-Way ANOVA | → Open |
| 🗂️ Categorical / Count Data | ||||
| Observed frequencies vs. an expected distribution | Roll a die 120 times — do the 6 outcomes appear equally often, as a fair die would predict? | Expected count ≥ 5 in each category | Chi-Square Goodness-of-Fit | → Open |
| Two categorical variables — are they related? | Is a customer's preferred product (A/B/C) associated with their gender? | Expected count ≥ 5 in each cell of the table | Chi-Square Independence | → Open |
| Proportion vs. a target value | A process should have a ≤5% defect rate. Out of 200 units, 14 were defective — is that significantly higher? | n·p₀ ≥ 5 and n·(1−p₀) ≥ 5 | Z-Test (one proportion) | → Open |
| Two independent proportions compared | Landing page A converts 12% of visitors, page B converts 15% — is that difference real or just noise? | n·p̂ ≥ 5 and n·(1−p̂) ≥ 5 in both groups | Two-Proportion Z-Test | → Open |
| ↔️ Comparing Variability (Spread) | ||||
| Equality of two variances | Machine A and Machine B both fill bottles to 500ml — is one machine more inconsistent (higher variance) than the other? | Both samples ≈ normally distributed | F-Test | → Open |
| 🔧 Non-Parametric (No Normality Required) | ||||
| 2 groups, skewed data / outliers / ordinal scale | Compare customer satisfaction ratings (1–5 scale) between two stores — ratings are ordinal, not normal. | Independent samples; no distribution shape assumed | Mann-Whitney U | → Open |
| 3+ groups, skewed data / outliers / ordinal scale | Compare repair times (heavily right-skewed) across 3 different technicians. | Independent samples; no distribution shape assumed | Kruskal-Wallis | → Open |