T-Test Calculator | One-Sample
Perform a one-sample t-test to determine if a sample mean differs significantly from a known value.
Test Type
Quick Examples
What Is the T-Test Calculator | One-Sample?
This sample t test calculator supports three types of hypothesis testing using the Student t distribution: one-sample (compare a sample mean to a known population value), two-sample Welch's t-test (compare two independent groups), and paired t-test (compare matched pairs like before/after measurements). Enter your data, define your null hypothesis (H₀), and click Calculate. The tool finds the t value, degrees of freedom, exact p-value, Cohen's d effect size, and 95% confidence interval.
- ›p-value computed via regularized incomplete beta function (exact, not table lookup)
- ›Degrees of freedom (df) calculated using Welch-Satterthwaite for two-sample tests, no equal-variance assumption
- ›Cohen's d effect size: small < 0.2, medium 0.2–0.8, large > 0.8
- ›95% CI shows the plausible range for the true mean difference
- ›Paired test requires equal-length datasets (matched subjects)
Formula
t-Test Formulas
One-Sample
t = (x̄ − μ₀) / (s/√n)
Two-Sample
t = (x̄₁−x̄₂) / √(s₁²/n₁+s₂²/n₂)
Paired
t = d̄ / (s_d/√n); d = x₁−x₂
Cohen's d
d = |x̄₁−x̄₂| / s_pooled
95% CI
x̄ ± t*(α/2) × SE
Welch df
(s₁²/n₁+s₂²/n₂)²/[(s₁²/n₁)²/(n₁−1)+…]
How to Use
- 1Select the test type: One-Sample, Two-Sample (Welch), or Paired
- 2Enter your data (numbers separated by spaces or commas)
- 3For one-sample: enter the null hypothesis value μ₀ (the population mean to test against)
- 4Set the significance level α (0.05 is standard) and choose one-tailed or two-tailed
- 5Click Calculate, the t value, degrees of freedom, and p-value appear instantly
- 6Read the hypothesis testing decision, confidence interval, and Cohen's d effect size with full step-by-step working
Example Calculation
One-sample t-test: Do heights differ from μ=170?
n=10, x̄=168.7, s=4.47, SE=1.414
t = (168.7 − 170) / 1.414 = −0.919
df = 9, p (two-tailed) = 0.382
→ Fail to reject H₀ (p > 0.05)
Two-sample: Group A (78,82,74,79,81) vs Group B (70,75,72,68,74)
SE = √(9.18/5 + 7.68/5) = 1.86
t = (78.8 − 71.8) / 1.86 = 3.76, df = 7
p = 0.007 → Reject H₀ (significant difference)
p-value vs Significance Level
In hypothesis testing, the p-value is the probability of observing a result at least as extreme as yours, assuming the null hypothesis (H₀) is true. It is NOT the probability that H₀ is true. A small p-value (e.g., p = 0.007) means your result is rare under H₀. Statistical significance (p < α) does not imply practical importance, always check Cohen's d for effect size.
Understanding T-Test | One-Sample
t-Test Quick Reference
| Test Type | Formula | df | Use When |
|---|---|---|---|
| One-Sample | t = (x̄−μ₀)/(s/√n) | n−1 | Compare mean to known value |
| Two-Sample (Welch) | t = (x̄₁−x̄₂)/SE_welch | Welch-Satt. | Compare two independent groups |
| Paired | t = d̄/(s_d/√n) | n−1 | Before/after same subjects |
| Cohen's d (small) | d < 0.2 | — | Negligible practical effect |
| Cohen's d (medium) | 0.5 ≤ d < 0.8 | — | Moderate practical effect |
| Cohen's d (large) | d ≥ 0.8 | — | Substantial practical effect |
Frequently Asked Questions
When should I use a one-sample vs two-sample t-test?
- ›One-sample: Is this batch's mean weight = 500g (manufacturer specification)?
- ›Two-sample: Do male and female students score differently on the same test?
- ›Paired: Does blood pressure decrease after this drug (measured on same patients)?
- ›Key distinction: paired removes between-subject variability, increasing power
Why use Welch's t-test instead of Student's t-test?
- ›Welch's df formula: (s₁²/n₁+s₂²/n₂)² / [(s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1)]
- ›If variances are equal, Welch gives nearly the same result as Student's t
- ›If variances differ (e.g., ratio 4:1), Student's t can be badly miscalibrated
- ›Modern recommendation: always use Welch's for two-sample tests
What is a p-value and how do I interpret it?
Common misconceptions about the p-value: it is NOT the probability the null hypothesis is true. It is NOT the probability of a false positive. It is NOT a measure of effect size.
- ›p < 0.05: conventional threshold (arbitrary; Neyman-Pearson framework)
- ›p < 0.01: "highly significant" in many fields
- ›Small sample + large p: may lack statistical power, not necessarily "no effect"
- ›Large sample: even trivial effects reach p < 0.05; check effect size
What is Cohen's d and how do I interpret it?
- ›d = 0.2: small effect (like the height difference between 15- and 16-year-olds)
- ›d = 0.5: medium effect (IQ difference between professional and non-professional occupations)
- ›d = 0.8: large effect (IQ gap between college and non-college graduates)
- ›A study can be statistically significant (low p) but practically unimportant (small d)
What assumptions does the t-test require?
- ›Normality: t-test is robust for n ≥ 30 by Central Limit Theorem
- ›Independence: each observation must be independent of the others
- ›Outliers can strongly distort the mean and t-statistic for small n
- ›Non-normal data with small n: consider Wilcoxon signed-rank (non-parametric alternative)
What is the 95% confidence interval for the mean difference?
- ›CI does not contain 0 ↔ two-tailed p < 0.05 (they are mathematically equivalent)
- ›Wider CI = more uncertainty (small n or high variability)
- ›Narrower CI = greater precision (large n or low variability)
- ›Report CIs rather than just p-values, they convey much more information
Is this t-test calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser.