T-Test Calculator

Perform statistical t-tests to compare means between groups. Calculate t-values, p-values, and determine statistical significance.

How to Use

1. Select test type (Independent or Paired)

2. Enter data for each group

3. Get comprehensive statistical results

Results Explained

T-Value: Test statistic

P-Value: Statistical significance

Mean Difference: Effect size

Enter numbers separated by commas or spaces

Enter numbers separated by commas or spaces

Theoretical Foundation

The t-test emerges from the fundamental principles of statistical inference, providing a rigorous framework for comparing means between populations. This statistical method, developed by William Sealy Gosset under the pseudonym "Student," addresses the challenges of small sample inference where population parameters are unknown. The test's theoretical foundation rests on the properties of the t-distribution, which naturally arises when estimating the mean of a normally distributed population.

The underlying theory accommodates the uncertainty in estimating population variance from sample data, making it more appropriate than the normal distribution for small sample analyses. The shape of the t-distribution, characterized by heavier tails than the normal distribution, reflects this additional uncertainty and approaches the normal distribution as sample size increases.

Mathematical Framework

The t-statistic calculation involves several key components:

Independent t-test:

t = (x̄₁ - x̄₂) / √(s²ₚ(1/n₁ + 1/n₂))

Where s²ₚ = ((n₁-1)s²₁ + (n₂-1)s²₂)/(n₁+n₂-2)

Paired t-test:

t = d̄ / (sd/√n)

Where:

  • x̄ = Sample means
  • s²ₚ = Pooled variance
  • d̄ = Mean difference
  • sd = Standard deviation of differences
  • n = Sample size

Distribution Properties

The t-distribution's probability density function is defined by:

f(t) = [Γ((v+1)/2)/(√(πv)Γ(v/2))] × (1 + t²/v)^(-(v+1)/2)

Where:

  • v = Degrees of freedom
  • Γ = Gamma function
  • π = Pi constant

The distribution's shape is determined solely by its degrees of freedom, which dictate the extent of its departure from normality. As degrees of freedom increase, the t-distribution converges to the standard normal distribution, reflecting reduced uncertainty in variance estimation.

Statistical Power Analysis

The power of a t-test depends on several interrelated factors: sample size, effect size, significance level (α), and the nature of the alternative hypothesis. The relationship between these factors can be expressed through the non-central t-distribution, which describes the sampling distribution under the alternative hypothesis. The power function for a two-sided test is given by:

Power = 1 - P(|T| ≤ t₁₋α/₂ | δ)

Where δ = (μ₁ - μ₂)/(σ/√n) = Effect size × √n

Computational Methods

The calculation of t-test probabilities involves sophisticated numerical methods for evaluating the cumulative distribution function of the t-distribution. Modern implementations typically use series expansions or continued fraction representations, often incorporating the relationship with the incomplete beta function:

P(T ≤ t) = 1 - ½I(v/(v+t²), v/2, 1/2)

Where:

  • I = Regularized incomplete beta function
  • v = Degrees of freedom

This computational approach ensures accurate probability calculations across the full range of degrees of freedom and test statistics, crucial for reliable statistical inference.