T-Statistic Calculator
Free t-statistic calculator. One-sample, two-sample pooled, Welch, paired t. p-value, critical value
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
T-Statistic Calculator โ Student's t when ฯ is Unknown
One-sample, two-sample (pooled/Welch), paired. Summary stats or raw data. p-value, critical value, Cohen's d effect size.
Real-World Scenarios โ Click to Load
Test Configuration
One-Sample Data
t-Distribution: Test Statistic and p-value Comparison
Effect Size vs Cohen's Benchmarks
Calculation Breakdown
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- One-sample: t = (xฬ โ ฮผโ) / (s/โn), df = nโ1. Compare sample mean to hypothesized value.
- Two-sample pooled: t = (xฬโโxฬโ) / (sโโ(1/nโ+1/nโ)), sโ = pooled SD, df = nโ+nโโ2.
- Welch's t: For unequal variances. df from Satterthwaite approximation.
- Paired: t = dฬ / (s_d/โn), df = nโ1. For before/after or matched pairs.
- p-value: From t-distribution CDF. Reject Hโ if p-value < ฮฑ.
- Cohen's d: Effect size. d = 0.2 small, 0.5 medium, 0.8 large.
- Critical value: t_ฮฑ,df or t_ฮฑ/2,df. Reject if |t| โฅ critical value.
Did You Know?
Formulas Reference
One-sample: t = (xฬ โ ฮผโ) / (s/โn), df = nโ1
Two-sample pooled: sโยฒ = [(nโโ1)sโยฒ+(nโโ1)sโยฒ]/(nโ+nโโ2)
Welch: t = (xฬโโxฬโ) / โ(sโยฒ/nโ + sโยฒ/nโ), df from Satterthwaite
Paired: t = dฬ / (s_d/โn), df = nโ1
Cohen's d (one-sample): (xฬ โ ฮผโ) / s
Cohen's d (two-sample): (xฬโโxฬโ) / s_pooled
When to Use Each Test
| Test | When to Use |
|---|---|
| One-sample | Single group mean vs hypothesized value (e.g., ฮผ = 100) |
| Two-sample pooled | Two independent groups, assume equal variances |
| Welch's t | Two independent groups, unequal or unknown variances |
| Paired | Same subjects measured twice (before/after) or matched pairs |
Frequently Asked Questions
When do I use pooled vs Welch?
Use pooled when you can assume equal variances (e.g., similar group sizes and SDs). Use Welch when variances may differ โ it's the default in many software packages.
What does the t-statistic mean?
t measures how many standard errors the observed difference is from the null. Larger |t| = stronger evidence against Hโ.
Why use t instead of z?
Use t when estimating ฯ from the sample (almost always). Use z only when population ฯ is known (rare).
What is effect size?
Cohen's d = (mean difference) / SD. It tells you the magnitude of the effect, not just whether it's significant.
How do I interpret p-value?
p-value = probability of observing your t (or more extreme) if Hโ were true. p less than 0.05 typically means reject Hโ.
Effect Size Scale (Cohen's d)
Interpreting the Chart
The blue curve is the t-distribution under Hโ. The red shaded area is the rejection region. The green shaded area is the p-value. If your t-statistic falls in the red region (or p-value < ฮฑ), reject Hโ.
Official Data Sources
Disclaimer: This calculator is for educational purposes. Verify assumptions (normality, equal variance) and use established statistical software for research.