T-Test Calculator
Run a one-sample, two-sample (Welch or pooled) or paired t-test from your data, with the t statistic, degrees of freedom, p-value, confidence interval and steps.
Results
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Steps
| Step | Working |
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Quick Answer
A t-test compares means. For two samples, t = (mean₁ − mean₂) ÷ SE. With the example data, sample means of 13.8125 and 12.1750 give Welch's t ≈ 4.0217. If the p-value is below 0.05, the difference is called statistically significant.
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Table of Contents
- T-Test Calculator
- Quick Answer
- Related Calculators
- How This T-Test Calculator Works
- How to Use the T-Test Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- T-Test vs Z-Test
- Two-Tailed Critical t Values
- Choosing the Right Test
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This T-Test Calculator Works
Choose the test, then paste your numbers. For two independent groups, use Welch's test (or the pooled test if the variances are equal). For before-and-after measurements on the same subjects, use the paired test. To compare one sample with a known value, use the one-sample test. The calculator finds the means, standard deviations, standard error, t statistic, degrees of freedom, p-value and a confidence interval for the difference.
Quick explanation: the t statistic measures the difference in standard-error units.
- Large |t|: the difference is large compared with the noise.
- Small p-value: such a difference would be unlikely if there were no real difference.
- Degrees of freedom: depend on sample sizes and set the shape of the t curve.
How to Use the T-Test Calculator
- Choose the type of t-test.
- Paste sample 1 and, if needed, sample 2.
- For a one-sample test, enter the hypothesized mean.
- Choose the tail and significance level.
- Press Calculate and read t, df, the p-value and the interval.
Core Formulas
- One-sample: t = (x̄ − μ₀) ÷ (s ÷ √n), df = n − 1
- Paired: t = d̄ ÷ (s_d ÷ √n), df = n − 1
- Welch: t = (x̄₁ − x̄₂) ÷ √(s₁²/n₁ + s₂²/n₂), with the Welch–Satterthwaite df
- Pooled: sp² = ((n₁ − 1)s₁² + (n₂ − 1)s₂²) ÷ (n₁ + n₂ − 2), df = n₁ + n₂ − 2
- Confidence interval: difference ± t* × SE
Key Terms
- t statistic: the difference divided by its standard error.
- Degrees of freedom: a number based on sample size that sets the t curve's shape.
- p-value: the chance of a result at least this extreme if the null hypothesis is true.
- Null hypothesis: the starting assumption, usually no difference.
- Standard error: the standard deviation of a sample mean or difference.
Worked Example
Test whether a mean of 52 from 16 scores with SD 8 differs from 48.
- SE = 8 ÷ √16 = 2
- t = (52 − 48) ÷ 2 = 2, with df = 15
- The two-tailed p-value is about 0.064, so at α = 0.05 the difference is not significant
Real-Life Examples
- A/B tests: compare average order values between two page designs.
- Medicine: compare blood pressure before and after a treatment with a paired test.
- Quality control: check whether parts match a target length with a one-sample test.
- School: compare test scores between two classes.
T-Test vs Z-Test
- T-test: used when the population standard deviation is unknown, which is most of the time.
- Z-test: used when the population standard deviation is known.
- Large samples: the t distribution gets very close to the normal curve.
Two-Tailed Critical t Values
Values from the t distribution.
| df | α = 0.10 | α = 0.05 | α = 0.01 |
|---|---|---|---|
| 1 | 6.314 | 12.706 | 63.657 |
| 5 | 2.015 | 2.571 | 4.032 |
| 10 | 1.812 | 2.228 | 3.169 |
| 15 | 1.753 | 2.131 | 2.947 |
| 20 | 1.725 | 2.086 | 2.845 |
| 30 | 1.697 | 2.042 | 2.750 |
| 60 | 1.671 | 2.000 | 2.660 |
| 120 | 1.658 | 1.980 | 2.617 |
Choosing the Right Test
- Different people in each group: two-sample test (Welch by default).
- Same people measured twice: paired test.
- One group against a target: one-sample test.
- Direction known in advance: a one-tailed test, decided before seeing the data.
- Very skewed data or outliers: consider a non-parametric test such as the Mann-Whitney U test instead.
Common Mistakes to Avoid
- Using a two-sample test on paired data. Paired data need the paired test.
- Choosing the tail after seeing the data. Decide before you look.
- Assuming equal variances. Welch's test is safer when unsure.
- Treating p > 0.05 as proof of no difference. It only means the evidence is not strong enough.
- Ignoring effect size. A tiny difference can be significant with huge samples.
Frequently Asked Questions
A test that checks whether a mean, or the difference between two means, is larger than chance would explain.
Two separate groups: Welch. Same subjects twice: paired. One group against a value: one-sample.
The chance of seeing a difference this large if there were really no difference.
A two-sample t-test that does not assume equal variances.
n − 1 for one-sample and paired tests, n₁ + n₂ − 2 for pooled, and the Welch–Satterthwaite formula for Welch.
There is no fixed minimum, but small samples need roughly normal data.
Yes, as numbers, but for proportions a z-test for proportions is often better.
A range of plausible values for the true mean difference.
By the usual rule, p must be below α to be significant.
ANOVA compares three or more groups. With two groups, it gives the same result as a pooled t-test.
Yes. Copy a column and paste it into the box.
From the exact t distribution using the incomplete beta function.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Student's t-test | One-sample, paired, pooled and Welch t-tests. | https://en.wikipedia.org/wiki/Student%27s_t-test |
| NIST/SEMATECH e-Handbook, Two-Sample t-Test for Equal Means | Formulas and assumptions. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm |