Updated Statistics

t-Test Calculator

Run one-sample, independent two-sample (Welch or pooled), and paired t-tests. Get t statistic, degrees of freedom, p-value (one- or two-tailed), confidence interval, effect size, and a decision at your chosen α.

One-sample Two-sample (Welch/Pooled) Paired CI + Effect Size
Note: Results assume valid t-test assumptions (independence, appropriate data type, and normality/CLT conditions). If data are heavily skewed or contain extreme outliers, consider robust or nonparametric alternatives.

t-Test Tool

Choose the test type tab, enter summary statistics (or paste data), and calculate. You can switch tail type, α, and confidence level any time.

One-sample t-test checks whether a sample mean differs from μ₀. If you paste data, the calculator will compute x̄, s, and n automatically.
Two-sample t-test compares group means. Welch is the safer default when variances differ. If you paste data for a group, its mean/SD/n are computed automatically.
A paired t-test is a one-sample t-test on the differences within pairs. Paste differences to compute d̄, s_d, and n automatically.
Common confidence levels Two-tailed α One-tailed α What it means
90% 0.10 0.05 Wider error tolerance; lower t*
95% 0.05 0.025 Most common default in practice
99% 0.01 0.005 Stricter threshold; higher t*
Critical t values come from the t distribution and depend on df. This tab helps you find the cutoff t* used in confidence intervals and hypothesis tests.

What a t-Test Is and Why It’s Used

A t-test is one of the most widely used statistical tests for answering a simple question: does a mean (or a difference in means) look “too far” from what we would expect if a null hypothesis were true? The t-test family is popular because it is easy to interpret, works well for moderate sample sizes, and remains useful even when you don’t know the population standard deviation.

In everyday analysis, t-tests show up whenever you’re comparing a sample average to a target benchmark (one-sample), comparing two groups (two-sample), or comparing before/after measurements on the same subjects (paired). This t-Test Calculator supports all three and reports the key outputs you usually need: t statistic, degrees of freedom, p-value, confidence interval, effect size, and a clear decision based on your chosen α.

t-Test Types Supported by This Calculator

Test type What it compares Typical use case Key idea
One-sample Sample mean vs μ₀ Is an average different from a target? Compare x̄ to μ₀ using s/√n
Two-sample (Welch) Two independent means Do two groups differ? Allows unequal variances (recommended)
Two-sample (pooled) Two independent means Groups with similar spread Assumes equal variances
Paired Mean of within-pair differences Before/after, matched pairs Run one-sample test on differences

The Core Formula Behind t-Tests

All t-tests boil down to the same structure: a difference (what you observed minus what the null hypothesis claims) divided by a standard error (how much that estimate typically varies). The resulting statistic follows a t distribution under the null hypothesis (assuming the test assumptions are reasonably met).

t = (estimate − null value) / standard error

One-sample t-test

The one-sample t-test checks whether a sample mean differs from a hypothesized population mean μ₀. It’s a standard choice when you have one set of measurements and a benchmark or target value.

t = (x̄ − μ₀) / (s / √n), df = n − 1

If you paste raw sample values into the calculator, it computes x̄ (mean), s (sample standard deviation), and n automatically. Otherwise, you can use summary statistics directly when you already have them.

Two-sample t-test: Welch vs pooled

A two-sample t-test compares the means of two independent groups. The difference is how we treat variances: Welch’s t-test does not assume equal variances and is typically the safest default, while the pooled (equal-variance) t-test combines variances into a single pooled estimate.

Welch: t = ((x̄₁ − x̄₂) − Δ₀) / √(s₁²/n₁ + s₂²/n₂)

Degrees of freedom for Welch are estimated using the Welch–Satterthwaite equation, which usually produces non-integer df. That’s normal and expected.

Pooled: s_p² = ((n₁−1)s₁² + (n₂−1)s₂²)/(n₁+n₂−2), t = ((x̄₁ − x̄₂) − Δ₀) / (s_p √(1/n₁ + 1/n₂)), df = n₁+n₂−2

Use pooled t-test only if you have reason to believe group variances are approximately equal (similar spreads, similar measurement processes). Otherwise, Welch is the more robust default and usually costs you little in power.

Paired t-test

A paired t-test is used when observations come in matched pairs: before/after measurements on the same subject, twin studies, matched case-control designs, repeated measures, and similar setups. Instead of treating the two sets as independent, you compute differences within each pair and test whether the mean difference is zero (or another hypothesized Δ₀).

Let dᵢ = (afterᵢ − beforeᵢ). Then t = (d̄ − Δ₀) / (s_d / √n), df = n − 1

This calculator lets you paste the differences directly. If you only have summary statistics for the differences, you can enter d̄, s_d, and n.

How to Interpret the p-Value (and Avoid Common Mistakes)

The p-value answers: “If the null hypothesis were true, how likely would we see a result at least as extreme as this one?” It does not directly tell you the probability the null hypothesis is true, and it does not measure the size or importance of an effect. It’s a measure of evidence against the null under the model assumptions.

This calculator supports two-tailed and one-tailed tests:

  • Two-tailed (≠): you care about differences in either direction (larger or smaller).
  • Right-tailed (>): you test whether the mean/difference is greater than the null value.
  • Left-tailed (<): you test whether the mean/difference is less than the null value.

Tail choice must be decided before looking at the data. Switching tails after inspecting results can inflate false positives.

Confidence Intervals (CI) and What They Tell You

A confidence interval gives a plausible range for the true mean (or mean difference) based on your sample. It’s often more informative than a binary “significant/not significant” decision because it shows both magnitude and uncertainty.

CI = estimate ± t* · SE

Where t* is the critical t value based on your confidence level and degrees of freedom, and SE is the standard error. If a two-tailed 95% CI for (μ₁−μ₂) does not include 0, the result corresponds to p ≤ 0.05 for the two-tailed test (same df and model).

Effect Size: Cohen’s d and Hedges’ g

Statistical significance depends strongly on sample size—small effects can become “significant” with large n. That’s why reporting an effect size helps: it standardizes the difference in a way that’s easier to compare across studies and contexts.

  • One-sample / paired: d ≈ (mean difference)/(SD). For paired, SD is s_d of the differences.
  • Two-sample: d ≈ (x̄₁ − x̄₂) / s_p (pooled SD) is common. For Welch, a pooled-style SD can still be used as a standardized scale; interpretation should note variance differences.
  • Hedges’ g: a small-sample correction of d, often used when n is small.

This tool reports an effect size estimate and is especially helpful when p-values are borderline or when you’re comparing multiple experiments.

Assumptions and Practical Checks

Like any statistical method, the t-test performs best when its assumptions are roughly satisfied:

  • Independence: observations within each group are independent (or differences are independent for paired).
  • Scale: data are continuous or approximately continuous and measured consistently.
  • Normality / CLT: the sampling distribution of the mean is approximately normal. With moderate sample sizes, the Central Limit Theorem often makes this reasonable, but extreme skew/outliers can still distort results.
  • Variance assumption (pooled only): pooled t-test assumes equal variances. If this is questionable, use Welch.

If your data contain heavy outliers, consider trimming, robust methods, or nonparametric alternatives (like the Wilcoxon signed-rank test for paired data or Mann–Whitney U for independent groups). If the data are binary or counts, a t-test may not be appropriate at all.

Step-by-Step: How to Use This t-Test Calculator

Quick workflow

  1. Select tail type (two-tailed or one-tailed) and choose α and confidence level.
  2. Pick the correct tab: One-Sample, Two-Sample, or Paired.
  3. Either enter summary stats (mean, SD, n) or paste raw data / differences.
  4. Click Calculate to get t, df, p-value, CI, effect size, and decision.
  5. If you need t*, use the Critical t tab (or read the critical t shown in each test output).

FAQ

t-Test Calculator FAQs

Common questions about choosing the right t-test, interpreting p-values, and reading confidence intervals.

A t-test is a statistical hypothesis test used to compare a sample mean to a known value, compare two independent sample means, or compare paired measurements, especially when the population standard deviation is unknown.

One-sample compares a sample mean to a hypothesized mean. Two-sample compares two independent group means (Welch or pooled). Paired compares the mean of within-pair differences (before/after or matched pairs).

Use Welch’s t-test when group variances may differ (recommended default). Use pooled t-test only when you can reasonably assume equal variances.

The p-value is the probability (under the null hypothesis) of observing a result at least as extreme as your data. A small p-value suggests evidence against the null hypothesis.

Alpha is your significance threshold, commonly 0.05. If p ≤ α, you reject the null hypothesis; if p > α, you fail to reject it.

A two-tailed test checks for differences in either direction. A one-tailed test checks for a difference in one specified direction only (greater-than or less-than).

Key assumptions include independent observations, approximately normal sampling distribution of the mean (often reasonable for moderate n), and for pooled two-sample t-test, equal variances.

Common options are Cohen’s d (standardized mean difference) and Hedges’ g (small-sample corrected). This calculator provides effect size estimates for each test type.

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