T-Test Calculator

Calculate one-sample, two-sample, and paired t-tests with p-values and confidence intervals

What is a T-Test?

A t-test is a statistical test used to determine if there is a significant difference between the means of two groups or between a sample mean and a known population mean. It is commonly used in hypothesis testing when the sample size is small and the population standard deviation is unknown.

Types of T-Tests

One-Sample T-Test:

Compares a sample mean to a known population mean

t = (x̄ - μ₀) / (s/√n)

Two-Sample T-Test:

Compares means of two independent groups

t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Paired T-Test:

Compares means of related groups (before/after)

t = d̄ / (s_d/√n)

T-Test Formula

The general formula for t-statistic is:

t = (difference) / (standard error)

Degrees of Freedom

  • One-sample: df = n - 1
  • Two-sample (equal variance): df = n₁ + n₂ - 2
  • Two-sample (unequal variance): df = (s₁²/n₁ + s₂²/n₂)² / ((s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1))
  • Paired: df = n - 1

How to Use the Calculator

  1. Choose the type of t-test you want to perform
  2. Enter your data (sample means, standard deviations, sample sizes)
  3. Set your significance level (α)
  4. Click "Calculate" to get the t-statistic and p-value
  5. Interpret the results

Interpreting Results

  • p-value < α: Reject the null hypothesis (significant difference)
  • p-value ≥ α: Fail to reject the null hypothesis (no significant difference)
  • |t| > critical value: Reject the null hypothesis
  • |t| ≤ critical value: Fail to reject the null hypothesis

Example Calculations

One-Sample T-Test:

Sample mean = 52

Population mean = 50

Sample SD = 5

Sample size = 25

t = (52 - 50) / (5/√25) = 2

Two-Sample T-Test:

Group 1 mean = 75, SD = 10, n = 30

Group 2 mean = 70, SD = 12, n = 30

t = (75 - 70) / √(10²/30 + 12²/30) = 1.67

Assumptions for T-Tests

  • Data should be normally distributed (approximately)
  • Observations should be independent
  • For two-sample tests, populations should have equal variances (unless using Welch's t-test)
  • Data should be continuous