A t-test is an inferential statistical test used to compare a sample mean with a known value or to compare two means. It uses the Student's t-distribution and is appropriate when the sample size is small or the population variance is unknown.

Which t-test should I use?

Use a one-sample t-test to compare a sample mean to a hypothesized mean, an independent two-sample t-test to compare two different groups, and a paired t-test to compare two measurements on the same subject (before/after).

What does the p-value tell me?

The p-value tells you how compatible your data are with the null hypothesis. A small p-value (commonly < 0.05) suggests the observed difference in means is unlikely to be due to random sampling variation alone.

t-test Calculator

Run a t-test online: one-sample, independent two-sample, or paired t-test. Enter your sample data or summary stats to get t-value, degrees of freedom, and p-value (one-tailed or two-tailed), plus formulas and interpretation.

Full original guide (expanded)

t-test Calculator

Choose the test type, enter sample statistics, and get t-value, degrees of freedom, and p-value (one- or two-tailed).

Sample mean

Sample SD

Sample size (n)

Hypothesized mean (μ₀)

Tail

Significance α

Used for interpretation only.

Results

t statistic

–

Degrees of freedom

–

p-value

–

How this t-test works

This calculator follows standard Student's t-test formulas.

1. One-sample t-test

Used to test whether a sample mean differs from a hypothesized population mean μ₀.

t = (x̄ − μ₀) / (s / √n)
df = n − 1

2. Independent two-sample t-test

Used to test whether two independent groups have different means.

Equal variances (pooled):

sₚ² = ((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2)
t = (x̄₁ − x̄₂) / ( sₚ √(1/n₁ + 1/n₂) )
df = n₁ + n₂ − 2

Welch (unequal variances):

t = (x̄₁ − x̄₂) / √( s₁²/n₁ + s₂²/n₂ )
df = ( s₁²/n₁ + s₂²/n₂ )² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]

3. Paired t-test

Used for before/after or matched-subject designs. Let d̄ be the mean of the differences and s_d their SD.

t = d̄ / (s_d / √n)
df = n − 1

Interpreting the p-value

If p < α (typically α = 0.05), you reject the null hypothesis and conclude that the means differ in the direction specified.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[','\\]

','\

Formula (extracted text)

t = (x̄ − μ₀) / (s / √n) df = n − 1

Formula (extracted text)

sₚ² = ((n₁ − 1)s₁² + (n₂ − 1)s₂²) / (n₁ + n₂ − 2) t = (x̄₁ − x̄₂) / ( sₚ √(1/n₁ + 1/n₂) ) df = n₁ + n₂ − 2

Formula (extracted text)

t = (x̄₁ − x̄₂) / √( s₁²/n₁ + s₂²/n₂ ) df = ( s₁²/n₁ + s₂²/n₂ )² / [ (s₁²/n₁)²/(n₁−1) + (s₂²/n₂)²/(n₂−1) ]

Formula (extracted text)

t = d̄ / (sd / √n) df = n − 1

Variables and units

PMI = private mortgage insurance (monthly) (currency)

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn

Formulas

(Formulas preserved from original page content, if present.)

Version 0.1.0-draft

Citations

Add authoritative sources relevant to this calculator (standards bodies, manuals, official docs).

Changelog

0.1.0-draft — 2026-01-19: Initial draft (review required).