Degrees of freedom. df = k − 1 = 6 − 1 = 5 (one fewer than the number of categories — that's the constraint that totals must sum to 60).
5
Look it up. Enter df = 5 and χ² = 2.6 above. Read the p-value: ≈ 0.76. Since p > 0.05, fail to reject H₀ — the data is consistent with a fair die.
Try it: change χ² to 12.0 with df=5 to see what a "loaded die" would look like (p ≈ 0.035, reject H₀).
Chi-Square Calculator
Enter values above — updates as you type.
Highlighted row = your df | Green cell = α bracket containing your test statistic | Reject H₀ when χ² exceeds the critical value for your chosen α
📚 Worked Example: One-sample z-test for a known population
The standardized SAT math section has population mean μ=500 and σ=100. A high school class of 36 students averaged 528. Did this class score significantly higher than the national average?
Compute z. z = (x̄ − μ) / (σ/√n) = (528 − 500) / (100/√36) = 28/16.67 ≈ 1.68.
3
Look it up. Enter z = 1.68 with the Upper tail selected. p ≈ 0.0465.
4
Conclude. Since p (0.0465) < α (0.05), reject H₀. This class did score significantly higher than the national average.
Why z and not t? Use z when you know the population σ. Use t when you only have the sample's standard deviation. With n=36 and a known σ, z is correct here.
Z-Score Probability Calculator
Enter a value and press Calculate — or just type.
Table shows P(Z ≤ z) — cumulative from left | Common critical values: 1.645 (α=0.05 one-tail) · 1.960 (α=0.025) · 2.326 (α=0.01 one-tail) · 2.576 (α=0.005)
📚 Worked Example: Two-sample t-test (independent groups)
You're comparing test scores between two teaching methods. Group A (n=20) had mean 78 with sd 8.5. Group B (n=22) had mean 84 with sd 7.9. Is method B better?
Pooled standard error. SE ≈ √(s²_A/n_A + s²_B/n_B) = √(72.25/20 + 62.41/22) = √(3.61 + 2.84) ≈ 2.54.
3
Compute t. t = (x̄_A − x̄_B) / SE = (78 − 84) / 2.54 ≈ −2.36.
4
Degrees of freedom. df ≈ n_A + n_B − 2 = 40 (Welch's correction may give slightly less; df=40 is fine for a quick check).
5
Look it up. Enter df = 40 and t = -2.36 with the Two-tailed option. p ≈ 0.023. Reject H₀ — method B produces significantly higher scores.
Effect size matters. Cohen's d ≈ 2|t|/√df ≈ 0.75 — a medium-to-large effect. Statistical significance alone doesn't tell you whether the difference is practically important; effect size does.
📚 Worked Example: Paired t-test (before/after)
You measure 15 patients' blood pressure before and after taking a new medication. The mean reduction is 6.2 mmHg with a sample sd of 8.4. Is the reduction significant?
1
Compute t. t = (mean_diff − 0) / (sd_diff/√n) = 6.2 / (8.4/√15) = 6.2/2.17 ≈ 2.86.
2
df. df = n − 1 = 14 (one fewer than the number of pairs).
3
Look it up. Enter df = 14, t = 2.86, Two-tailed → p ≈ 0.013. The medication produces a significant reduction.
Why paired and not independent? Each subject is their own control — you remove between-subject variability and gain power. Always pair when measurements are linked.
t-Student Calculator
Enter values above — updates as you type.
At df=∞ the t-distribution equals the standard normal | Highlighted row = your df | Green cell = α bracket for your test statistic
📚 Worked Example: One-way ANOVA
You're comparing the effectiveness of three teaching methods. Three groups of 15 students each (n=45 total) take the same test. Group means: A=72, B=78, C=81. The ANOVA gives F = 4.21.
1
Hypothesis. H₀: μ_A = μ_B = μ_C (all equal). H₁: at least one differs.
2
Degrees of freedom. df₁ = k − 1 = 3 − 1 = 2 (numerator, between-group). df₂ = N − k = 45 − 3 = 42 (denominator, within-group).
3
Look it up. Enter df₁ = 2, df₂ = 42, F = 4.21. p ≈ 0.022. Reject H₀ — at least one method differs.
4
Effect size. η² = (df₁·F) / (df₁·F + df₂) = 8.42 / 50.42 ≈ 0.167. By Cohen's conventions this is a large effect — about 17% of the variance in test scores is explained by teaching method.
5
Post-hoc. ANOVA only tells you "at least one differs" — to know which, run Tukey HSD or Games-Howell on the Corrections tab.
F-test isn't just for ANOVA. It's also the variance ratio for testing whether two populations have equal variances (Levene's, Bartlett's), and the joint significance test for regression coefficients. Same distribution, different applications.
F-Distribution Calculator
Enter values above — updates as you type.
Table shows critical F-values for the selected α | Rows = df₂ (denominator) | Columns = df₁ (numerator) | Reject H₀ when F exceeds the critical value | Used in ANOVA and regression F-tests
📚 Worked Example: Finding the critical value for any α
You're running a new chi-square goodness-of-fit test with df=8. Before collecting data, you want to know: how large does my χ² statistic need to be to reach significance at α = 0.01?
1
Pick the distribution. Chi-square (χ²).
2
Set α. α = 0.01 (1% significance level).
3
Set tail. Upper-tailed (chi-square is always upper-tailed for goodness-of-fit).
4
Set df. df = 8.
5
Read the result. Critical value ≈ 20.09. If your computed χ² exceeds this, reject H₀.
Use this for power planning. Knowing the critical value before running an experiment helps you decide if your design has enough sample size to detect a realistic effect.
Enter parameters above to get the exact critical value, rejection region, and decision rule.
2×2 Contingency Table
Column 1
Column 2
Row 1
Row 2
Paste raw numbers (one per line or comma-separated). For ANOVA, separate groups with blank lines. For chi-square independence, paste a contingency table (rows on lines, values comma-separated). For multiple regression, each row in Sample 1 should be x₁,x₂,…,xₖ. All computation is client-side.
Power = probability of correctly rejecting H₀ when the effect is real | Cohen's conventions: small d=0.2, medium d=0.5, large d=0.8 | Chi-square uses w: small=0.1, medium=0.3, large=0.5
Mann-Whitney U: non-parametric alternative to two-sample t-test | Reject H₀ when U ≤ critical value | No normality assumption required
At least 3 paired values required
Must have same count as Variable X
Enter paired values for X and Y to compute Pearson's correlation coefficient (r), R², t-statistic, and p-value with full intermediate calculations. All computation is client-side.
Binomial distribution: models the number of successes in n independent Bernoulli trials, each with probability p.
Paste at least 2 p-values to see all 5 correction methods.
Family-wise methods (Bonferroni, Holm, Hochberg) control the probability of any false positive; FDR methods (Benjamini-Hochberg, Benjamini-Yekutieli) control the expected proportion of false positives among rejections. Holm strictly dominates Bonferroni; Hochberg is uniformly at least as powerful as Holm when assumptions hold. BH-FDR is standard in genomics and large-scale testing.
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A Practical Guide to the Statistical Tests on This Page
Every test on this page exists because researchers needed an answer to a specific kind of question. This guide walks through what each one does, when to use it, and how to read the result — written for students and practitioners, not statisticians.
Choosing the right test
The first question isn't which test — it's what kind of data do you have?
Comparing one proportion or mean to a known value → one-sample z-test (known σ) or t-test (sample sd).
Comparing two group means → two-sample t-test (independent or paired, depending on data structure).
Comparing 3+ group means → one-way ANOVA (F-test), then post-hoc tests (Tukey HSD, Games-Howell).
Counts in categories → chi-square (goodness-of-fit or contingency).
Linear relationship between two continuous variables → Pearson correlation or OLS regression.
Distribution-free / ordinal data / small samples → Mann-Whitney U, Wilcoxon, Kruskal-Wallis.
Multiple tests on the same data → adjust p-values with Bonferroni, Holm, or Benjamini-Hochberg FDR.
Z-distribution (standard normal)
Use when you know the population standard deviation σ, or when your sample is large (n ≥ 30) and you can treat the sample sd as a population sd.
Formula:z = (x − μ) / σ
For a sample mean: z = (x̄ − μ) / (σ/√n)
Reading the result: if the p-value is below your chosen α (commonly 0.05), reject the null hypothesis. Two-tailed tests double the area; one-tailed tests use a single side.
Common use cases: SAT scores, IQ scores, manufacturing tolerances, large-sample proportion tests, control-chart limits.
t-distribution (Student's t)
Use when σ is unknown and the sample is small. The t-distribution has heavier tails than the normal — accounting for the extra uncertainty from estimating σ from the sample.
One-sample formula:t = (x̄ − μ) / (s/√n)
Two-sample (independent):t = (x̄₁ − x̄₂) / SE where SE pools the group variances.
Paired:t = mean_diff / (sd_diff/√n)
Effect size (Cohen's d): 0.2 = small, 0.5 = medium, 0.8 = large. Always report d alongside p — significance without effect size is a partial answer.
Convergence note: at df ≥ 30, t and z give nearly identical results. Below that, use t.
Chi-square (χ²)
Use for categorical data — counts in cells. Two main flavors:
Goodness-of-fit: do my observed counts match a theoretical distribution? χ² = Σ (O − E)² / E with df = k − 1.
Test of independence: are two categorical variables related? Build a contingency table; df = (rows−1)(cols−1).
Effect size: Cramer's V V = √(χ²/(N·min(r−1,c−1))). 0.1 / 0.3 / 0.5 = small / medium / large.
Caution: chi-square requires expected counts ≥ 5 in each cell. Below that, use Fisher's exact test instead.
F-distribution and ANOVA
The F-statistic is a ratio of variances. In one-way ANOVA: F = MSB / MSW where MSB is between-group mean square and MSW is within-group mean square.
Two degrees of freedom: df₁ = k − 1 (numerator, between-group); df₂ = N − k (denominator, within-group). Don't mix them up.
Effect size: η² = (df₁·F) / (df₁·F + df₂). 0.01 / 0.06 / 0.14 = small / medium / large.
What ANOVA does NOT tell you: if you reject H₀, you only know at least one group differs. To find which, run Tukey HSD (equal variances) or Games-Howell (unequal).
Other uses: joint significance test for regression coefficients; Levene's test for equal variances; ratio of two variances.
Correlation and Regression
Pearson r measures linear association between two continuous variables. Range: −1 to +1. r = Cov(X,Y) / (σ_X · σ_Y)
Spearman ρ uses ranks instead of raw values — robust to outliers and non-linear monotonic relationships.
r² (coefficient of determination): proportion of variance in Y explained by X. 0.01 / 0.09 / 0.25 = small / medium / large.
OLS regression assumptions: linearity, independence (Durbin-Watson), homoscedasticity (residual plot), normality of residuals (Q-Q plot), no multicollinearity (VIF). The Correlation tab on this page checks all of these.
Non-parametric tests
Use when your data violates t-test/ANOVA assumptions: small sample, ordinal data, non-normal distribution, presence of outliers.
Mann-Whitney U: two independent groups, ordinal/non-normal data. Tests whether one distribution stochastically dominates the other.
Wilcoxon signed-rank: paired samples, ordinal data. The non-parametric paired t-test.
Kruskal-Wallis H: 3+ independent groups. The non-parametric one-way ANOVA. Use Dunn's test for post-hoc.
Shapiro-Wilk / Anderson-Darling / Jarque-Bera: tests of normality. Reject H₀ → use a non-parametric test.
Power analysis and sample size
Statistical power = 1 − β = the probability of correctly rejecting a false null. Conventional target: 0.80 (80%).
Inputs: α (significance), effect size, sample size. Specify any three; solve for the fourth.
Why it matters: too small a sample = underpowered study (you'll miss real effects). Too large = wasted resources and risk of detecting trivial effects as "significant".
Use the Power tab on this page to find the minimum n needed to detect a realistic effect at your desired power.
Multiple comparison corrections
Running 20 tests at α = 0.05 means you'd expect 1 false positive purely by chance. Three corrections protect against this:
Bonferroni: divide α by the number of tests. Strictly conservative — controls family-wise error rate (FWER) but loses power.
Holm-Bonferroni: step-down procedure. Same FWER protection as Bonferroni but more powerful.
Benjamini-Hochberg FDR: controls the false-discovery rate (proportion of rejections that are false). Standard in genomics, neuroimaging, and any high-throughput screening. Most powerful — use this for exploratory work.
Tukey HSD: ANOVA post-hoc, equal variances and equal n.
Games-Howell: ANOVA post-hoc, unequal variances or unequal n.
Reading p-values correctly (the most common mistake in stats)
A p-value is the probability of obtaining a result at least as extreme as observed, assuming the null hypothesis is true. It is not:
The probability that the null hypothesis is true. (That's a Bayesian quantity that requires a prior.)
The probability that your hypothesis is correct.
An effect-size measure. A tiny effect can produce p < .001 with a large enough sample; a huge effect can produce p > .05 with a small sample.
Evidence that an effect "doesn't exist" if p > .05. Failure to reject is not the same as "accepting" the null.
The American Statistical Association's 2016 statement on p-values (The ASA Statement on p-Values: Context, Process, and Purpose) is the canonical reference if you want to go deeper.
APA-style reporting (what to write in your paper)
For a t-test: t(df) = t-value, p = .xxx, d = x.xx — for example, t(40) = 2.36, p = .023, d = 0.75.
For ANOVA: F(df1, df2) = F-value, p = .xxx, η² = .xx — e.g. F(2, 42) = 4.21, p = .022, η² = .17.
For correlation: r(df) = r-value, p = .xxx — e.g. r(28) = .42, p = .021.
For chi-square: χ²(df, N = n) = χ²-value, p = .xxx, V = .xx — e.g. χ²(2, N = 200) = 11.42, p = .003, V = .24.
Always report exact p-values to three decimals (or p < .001 when smaller). Include the effect size — APA 7th edition treats it as required, not optional.
Privacy & data handling
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Frequently asked questions
Why does this give a slightly different p-value than R / SPSS / Python?
Numerical approximations of the incomplete gamma, regularized incomplete beta, and Studentized range distributions converge at different rates depending on the algorithm. Our values agree with R, scipy, and SPSS to 3–4 decimal places — accurate enough for any APA-style reporting and for hypothesis testing at conventional α levels. If you need bit-exact agreement for forensic or regulatory work, use R or scipy as your primary tool.
Why isn't there a Bayesian option?
Bayesian inference requires choosing priors, which is genuinely subject-matter-specific. JASP and brms are excellent free Bayesian tools that handle this correctly. We may add Bayes factor calculators in a future version, but doing it half-right (with a default prior the user doesn't understand) would be worse than not doing it at all.
Can I use this for my thesis / dissertation / publication?
Yes — the numerical methods are accurate to 3–4 decimal places, which exceeds APA reporting requirements. Cite the original tests and corrections (e.g. "Tukey, 1949"; "Benjamini & Hochberg, 1995") in your methods section, not this tool. We're a calculator, not a primary source.
What about effect-size calculators for tests not shown?
The Correlation tab gives you r and r² directly. The t-test panel shows Cohen's d. The F-test panel shows η². For chi-square, divide χ² by N and take the square root for ϕ (2×2 tables) or use Cramer's V (formula above). For non-parametric tests, the rank-biserial correlation is a good general effect size. We're working on adding all of these as first-class outputs.