Student’s t-test Calculator
Run one-sample, Welch or pooled independent, and paired t-tests from observations or summaries; inspect the mean difference, SE, p-value and interval.
Choose by data structure · Welch default · pre-specify H₁ and α
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
The t statistic measures the estimated departure from the null in standard errors. With the same mean gap, greater dispersion usually raises SE; more independent observations usually lower it. A p-value is the probability of a statistic at least as extreme under H₀ and the model assumptions, not the probability that H₀ is true.
See calculation method and assumptions
Formula
One-sample and paired differences
Welch mean difference and SE
Pooled equal variance and two-sided interval
Variables, rates and periods
- H₀ / H₁
- Null and alternative; choose direction before inspecting data.
- x̄, μ₀, s, n
- Sample mean, null mean, sample SD and count.
- vA, vB, ν
- sA²/nA, sB²/nB and Welch degrees of freedom, which can be fractional.
- sp², SE
- Pooled variance and standard error; pooling assumes equal variances.
- dᵢ, δ₀
- Paired A−B difference and null difference.
- α, p
- Preset significance level and extremeness probability under the null model.
Define the question first
- For one sample, H₀ specifies a population mean μ₀.
- Independent and paired tests usually specify a mean difference Δ₀.
- H₁ may be unequal, greater or less.
- Choose α, the planned Type I error rate, and the direction before examining the outcome.
SD, SE and effect size
- Sample SD describes individual dispersion;
- SE describes sampling variation in the estimate.
- A tiny mean difference can become significant in a large sample while remaining unimportant.
- Ask how large the difference is, how uncertain it is and whether it changes a decision.
Independent and paired observations
- Two groups of different units need an independent-sample method.
- Before–after observations on the same units need aligned differences.
- Handle missing pairs together;
- independently removing blanks and pairing the remaining rows changes the data.
Conditions for t inference
- Observations or pair differences should be representative and independent.
- Small samples need approximately normal errors without severe outliers.
- Larger samples do not automatically fix selection bias, serial dependence or heavy tails.
- More trading days are not necessarily more independent information.
Run one-sample, Welch or pooled independent, and paired t-tests from observations or summaries; inspect the mean difference, SE, p-value and interval.
Worked example
- Choose one-sample and enter data or n=20, mean=53 and sample SD=6.
- Set μ₀=50 and α=0.05; choose the alternative before looking at results.
- Read the estimated difference, SE, df, p-value and matching interval, then check sampling assumptions.
Frequently asked questions
Why is Welch the independent-sample default?
Welch allows unequal population variances and uses a separate SE contribution from each group with approximate df. Pooled t adds an equal-variance assumption; equal sample sizes alone do not justify it.
How does pairing change the test?
A paired test is a one-sample t-test of within-unit differences. Shared individual variation may cancel, but the differences still need independence and suitable distribution assumptions.
Why is a one-tail interval one-sided?
A directional alternative has a one-sided confidence bound and an unbounded opposite end. The tool labels that end as unbounded rather than inventing a finite number. Two-sided tests have two-sided intervals.
Related calculators
Mean Confidence Interval Calculator
Estimate a mean interval using Student’s t.
Statistical Distributions & Critical Values Calculator
Explore normal, t, χ² and F tails and critical values.
One-way ANOVA Calculator
Separate between/within variation and the overall test.