t-Value Calculator

Perform t-tests, calculate t-values, and determine statistical significance. One-sample, two-sample, and paired t-test analysis.

One-Sample t-Test
Two-Sample t-Test
Paired t-Test

One-Sample t-Test: Compares sample mean to a known population mean

Formula: t = (x̄ - μ) / (s/√n) where x̄ is sample mean, μ is population mean, s is sample SD, n is sample size

Sample Data
Enter your sample data values

Example Data Sets:

Test Scores
Product Weights
Response Times
Test Parameters
The known or hypothesized population mean
Direction of the alternative hypothesis

Two-Sample t-Test: Compares means of two independent samples

Formula varies based on equal/unequal variances assumption

Group 1 Data
Enter data for the first group
Group 2 Data
Enter data for the second group
Test Parameters
Direction of the alternative hypothesis
Assumption about population variances

Example Data Sets:

Control vs Treatment
Group A vs Group B

Paired t-Test: Compares means of two related samples (before/after, matched pairs)

Formula: t = d̄ / (s_d/√n) where d̄ is mean difference, s_d is SD of differences

Paired Data
Enter data for before treatment or first measurement
Enter data for after treatment or second measurement

Example Data Sets:

Before/After Treatment
Pre/Post Test
Test Parameters
Direction of the alternative hypothesis
Significance Level (α):
α = 0.05 (Standard)
α = 0.01 (Strict)
α = 0.10 (Liberal)
Custom α
Calculating t-value and performing analysis...

Understanding t-Tests

The t-test is a statistical test used to compare means and determine if differences between groups are statistically significant. It's commonly used when sample sizes are small and population variance is unknown.

Student's t-Distribution:

The t-distribution is similar to the normal distribution but has heavier tails. As sample size increases, the t-distribution approaches the normal distribution.

Types of t-Tests

One-Sample
  • Compares sample mean to known value
  • t = (x̄ - μ) / (s/√n)
  • df = n - 1
Two-Sample
  • Compares two independent groups
  • Equal or unequal variances
  • df = n₁ + n₂ - 2 (equal var)
Paired
  • Compares related measurements
  • Before/after, matched pairs
  • df = n - 1

Assumptions of t-Tests

1

Normality: Data should be approximately normally distributed. The t-test is robust to moderate violations of this assumption, especially with larger sample sizes.

2

Independence: Observations should be independent of each other. For paired tests, the pairs should be independent but measurements within pairs are related.

3

Homogeneity of Variance: For two-sample t-tests with equal variances assumption, the two groups should have similar variances. If not, use Welch's t-test (unequal variances).

4

Random Sampling: Data should come from a random sample from the population of interest.

When to Use Each Test

Situation Appropriate Test Example
Compare sample to known value One-sample t-test Is average test score different from 75?
Compare two independent groups Two-sample t-test Do treatment and control groups differ?
Compare related measurements Paired t-test Before vs after treatment measurements
Variances unknown but assumed equal Equal variances t-test Comparing similar populations
Variances unknown and unequal Welch's t-test Comparing different populations

Calculator Features:

  • Performs one-sample, two-sample, and paired t-tests
  • Calculates t-statistics, p-values, and degrees of freedom
  • Provides confidence intervals and effect sizes
  • Visualizes t-distribution and critical regions
  • Includes detailed step-by-step calculations

Frequently Asked Questions

Use a z-test when population variance is known or sample size is large (n ≥ 30). Use a t-test when population variance is unknown and sample size is small (n < 30). The t-distribution has heavier tails than the normal distribution, accounting for additional uncertainty when estimating population variance from a sample.

Use an F-test or Levene's test to check if variances are equal. If the test indicates unequal variances (p < 0.05), use Welch's t-test (unequal variances). When in doubt, use Welch's test as it's more robust and doesn't assume equal variances. The equal variances test has slightly more power when variances are actually equal.

The t-test can be used with very small samples (as low as n = 2 for paired tests, n = 3 per group for independent tests). However, small samples have low power to detect effects unless they're very large. For 80% power to detect a medium effect (Cohen's d = 0.5) at α = 0.05, you need about 64 participants per group for a two-sample test.

Cohen's d is a measure of effect size. For a one-sample test: d = (x̄ - μ) / s. For a two-sample test with equal variances: d = (x̄₁ - x̄₂) / s_pooled. For paired tests: d = d̄ / s_d. Interpretation: d = 0.2 (small), d = 0.5 (medium), d = 0.8 (large). Effect size helps interpret practical significance beyond statistical significance.

The t-test is reasonably robust to violations of normality, especially with larger sample sizes (n > 30) due to the Central Limit Theorem. For severely non-normal data or small samples, consider non-parametric alternatives: Mann-Whitney U test (independent samples) or Wilcoxon signed-rank test (paired samples). Always check normality with Q-Q plots or Shapiro-Wilk test if concerned.