T-Test Calculator

The t-test is a statistical test used to determine whether there is a meaningful difference between group means. This calculator supports both one-sample and two-sample (Welch’s) t-tests, computing the t-statistic, degrees of freedom, and p-value.

One-Sample T-Test:

t = (x̄ - μ₀) / (s / √n)

Tests whether a sample mean differs from a known or hypothesized value μ₀.

Two-Sample T-Test (Welch’s, independent):

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Tests whether two groups have different means, without assuming equal variances.

What each variable means:

• x̄: the sample mean (average of your data values).
• μ₀: the hypothesized population mean you are testing against.
• s: the sample standard deviation, measuring spread in your data.
• n: the number of observations in the sample.
• s₁², s₂²: the variances of sample 1 and sample 2, respectively.

When to use a t-test: Use it when comparing means and the population standard deviation is unknown (which is almost always the case in practice). Use a one-sample test to check if a group differs from a target value, and a two-sample test to compare two groups.

Practical example: A one-sample test with data [5.1, 4.9, 5.3, 5.0, 4.8] against a hypothesized mean of 5.0. The sample mean = 5.02, SD = 0.192, SE = 0.086. t = (5.02 - 5.0) / 0.086 = 0.233 with 4 degrees of freedom. The p-value is 0.827, which is not significant at 0.05 — so we cannot conclude the mean differs from 5.0.

Interpreting results:

• If the two-tailed p-value is less than 0.05, the difference is statistically significant.
• A larger absolute value of t means a bigger difference relative to variability.
• Degrees of freedom for the two-sample test use Welch’s approximation, which adjusts for unequal variances.

Common mistakes: Make sure your data is roughly normally distributed, especially for small samples. The t-test compares means — it does not test whether distributions are the same shape. Use a chi-square test for categorical data instead.