Calculator
Pick a test, then enter summary values or paste raw data. Fields not used by your selected test can be left blank.
Example data table
These examples show typical input patterns and expected interpretation.
| Scenario | Inputs | Output highlights |
|---|---|---|
| One-sample mean (t) | x̄=52, s=10, n=25, μ₀=50, α=0.05, two-tailed | t ≈ 1.0, p-value ≈ 0.326, fail to reject H₀ |
| One-proportion | x=62, n=100, p₀=0.50, α=0.05, right-tailed | z ≈ 2.4, p-value ≈ 0.008, reject H₀ |
| Two-sample means (Welch) | x̄₁=5.1 s₁=1.2 n₁=20; x̄₂=4.5 s₂=1.0 n₂=22 | t, df≈, CI for (μ₁−μ₂), decision at α |
| Paired t-test | raw lists: before/after scores, same length | t on differences, CI for mean change |
| Two-proportion | x₁=30 n₁=80; x₂=18 n₂=75; α=0.05 | z, p-value, CI for p₁−p₂ |
Numbers above are illustrative and may differ slightly from your results due to rounding.
Formulas used
Mean tests
Z = (x̄ − μ₀) / (σ / √n)
t = (x̄ − μ₀) / (s / √n)
Two-sample (Welch): t = ((x̄₁−x̄₂)−Δ₀)/√(s₁²/n₁ + s₂²/n₂)
Pooled (equal variances): use s_p² and df = n₁+n₂−2.
Proportion tests
z = (p̂ − p₀) / √(p₀(1−p₀)/n)
z = ((p̂₁−p̂₂)−Δ₀) / SE
For Δ₀=0, SE uses pooled p̂ = (x₁+x₂)/(n₁+n₂).
p-value comes from the chosen tail and the CDF.
Confidence intervals
Mean CI: estimate ± critical · SE (critical from Z or t). Proportion CI: p̂ ± z_{α/2}·√(p̂(1−p̂)/n). Two-sample CI uses SE for the difference.
How to use this calculator
- Select the test type that matches your question.
- Choose the tail that matches your alternative hypothesis.
- Set α, then enter summary stats or paste raw data.
- Press Compute to view results above the form.
- Use the p-value and decision line to conclude.
- Download CSV or PDF to save your report.
FAQs
1) Which tail should I choose?
Use two-tailed for “different”. Use right-tailed for “greater”. Use left-tailed for “less”. Tail choice must match your planned hypothesis.
2) When should I use Z instead of t?
Use Z for means only when the population standard deviation σ is truly known. Otherwise, t is safer because it accounts for estimating variability from the sample.
3) What does the p-value mean here?
It is the probability, under the null hypothesis, of getting a statistic at least as extreme as yours. Smaller p-values indicate stronger evidence against H₀.
4) Why does Welch differ from pooled t?
Welch does not assume equal variances and uses an adjusted degrees-of-freedom. It is more robust when group spreads differ or when sample sizes are unbalanced.
5) Can I paste raw data instead of summary values?
Yes. If you provide raw data list 1 (and list 2 when needed), the calculator computes mean, standard deviation, and sample size automatically and uses those values.
6) Why might my p-value differ from software?
Small differences can come from rounding and numerical approximations for the t distribution. For high-stakes analysis, confirm with a trusted statistics package.
7) Is this valid for very small samples?
t-tests can work with small samples if assumptions are reasonable. For proportions with small n or extreme p, exact tests may be better than normal approximations.