T-Distribution Calculator

Calculate probabilities and critical values for the t-distribution with interactive visualization

What is the T-Distribution?

The t-distribution (Student's t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population with unknown variance from a small sample. It's wider than the normal distribution and approaches it as sample size increases.

PDF Formula: f(t) = Γ((ν+1)/2) / (√(νπ) × Γ(ν/2)) × (1 + t²/ν)^(-(ν+1)/2)

Where ν (nu) is the degrees of freedom and Γ is the gamma function

Common Applications:

• Hypothesis testing with small samples
• Confidence intervals for means
• Regression analysis
• Quality control testing
• Medical research studies

Interactive Calculator

Degrees of Freedom (ν)

Usually n-1 where n is sample size

T-Value

The t-statistic value

Quick Results

Enter values and click calculate to see results

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Detailed Statistical Solutions

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Distribution Visualization

T-Distribution Properties

Mean (μ)

μ = 0 (for ν > 1)

Variance (σ²)

σ² = ν/(ν-2)

Std Dev (σ)

σ = √(ν/(ν-2))

Degrees of Freedom

Skewness

0 (for ν > 3)

Excess Kurtosis

6/(ν-4)

Mode

Always 0

Key Statistical Properties

Fundamental Rules:

• Symmetric around zero
• Heavier tails than normal distribution
• Approaches normal as ν → ∞
• Variance > 1 for finite degrees of freedom

Practical Guidelines:

• Use when σ is unknown and n < 30
• For ν ≥ 30, approximately normal
• Critical for small sample hypothesis tests
• Essential for confidence intervals

Complete Guide to the T-Distribution

Introduction

The t-distribution, also known as Student's t-distribution, is one of the most important continuous probability distributions in inferential statistics. Developed by William Sealy Gosset (who published under the pseudonym "Student") in 1908, this distribution is essential for statistical inference when dealing with small sample sizes and unknown population variance.

Unlike the standard normal distribution, the t-distribution has heavier tails and is characterized by a single parameter: the degrees of freedom (ν). As the degrees of freedom increase, the t-distribution approaches the standard normal distribution, making it a versatile tool for various statistical applications.

Mathematical Foundation

Probability Density Function

The PDF of the t-distribution is given by:

f(t) = Γ((ν+1)/2) / (√(νπ) × Γ(ν/2)) × (1 + t²/ν)^(-(ν+1)/2)

Where ν > 0 is the degrees of freedom and Γ is the gamma function

Key Properties

• Mean (μ) = 0 (for ν > 1)
• Variance (σ²) = ν/(ν-2) (for ν > 2)
• Standard Deviation (σ) = √(ν/(ν-2))
• Mode = 0
• Skewness = 0 (for ν > 3)

Distribution Characteristics

• Continuous probability distribution
• Symmetric around zero
• Bell-shaped with heavier tails
• Support: (-∞, +∞)
• Approaches N(0,1) as ν → ∞

When to Use the T-Distribution

Required Conditions

The t-distribution is appropriate when:

Small Sample Size: Typically n < 30
Unknown Population Variance: σ² is unknown and estimated by s²
Normal Population: The underlying population is approximately normal
Independent Observations: Sample observations are independent

❌ Not Suitable For

• Large samples (n ≥ 30) with known σ
• Highly skewed populations
• Dependent observations
• Non-normal populations (small n)

✅ Perfect For

• Small sample hypothesis tests
• Confidence intervals for μ
• Regression analysis
• Quality control (small batches)

⚠️ Consider Alternatives

• Z-test (large samples, known σ)
• Wilcoxon test (non-normal data)
• Bootstrap methods
• Chi-square test (variance testing)

Real-World Applications

Medical Research

Clinical Trials

Testing drug effectiveness with small patient groups, comparing treatment outcomes.

Laboratory Analysis

Analyzing blood test results, enzyme levels, or biomarker concentrations.

Epidemiology

Studying disease prevalence in small populations or specific demographics.

Business and Quality Control

Manufacturing

Quality control testing with small batch sizes, process capability studies.

Market Research

Consumer preference studies, A/B testing with limited sample sizes.

Finance

Portfolio performance analysis, risk assessment with limited historical data.

Worked Examples

Example 1: One-Sample t-Test

Problem: A manufacturer claims their light bulbs last 1000 hours on average. A sample of 12 bulbs shows a mean life of 950 hours with a standard deviation of 80 hours. Test at α = 0.05.

Solution:

H₀: μ = 1000, H₁: μ ≠ 1000

Test statistic: t = (x̄ - μ₀)/(s/√n) = (950 - 1000)/(80/√12) = -2.165

Degrees of freedom: df = n - 1 = 11

Critical value: t₀.₀₂₅,₁₁ = ±2.201

Decision: |t| = 2.165 < 2.201, fail to reject H₀

Conclusion: Insufficient evidence to reject the manufacturer's claim at α = 0.05.

Example 2: Confidence Interval

Problem: A sample of 8 measurements has x̄ = 25.3 and s = 3.2. Construct a 95% confidence interval for the population mean.

Solution:

Sample statistics: n = 8, x̄ = 25.3, s = 3.2

Degrees of freedom: df = 8 - 1 = 7

Critical value: t₀.₀₂₅,₇ = 2.365

Margin of error: E = t × (s/√n) = 2.365 × (3.2/√8) = 2.677

Confidence interval: 25.3 ± 2.677 = [22.623, 27.977]

Interpretation: We are 95% confident that the true population mean lies between 22.623 and 27.977.

Relationship to Other Distributions

Standard Normal Distribution

As ν → ∞, the t-distribution converges to the standard normal distribution N(0,1). For practical purposes, when ν ≥ 30, the difference is negligible.

Rule of thumb: Use Z-distribution when n ≥ 30 or σ is known

Chi-Square Distribution

If T ~ t(ν), then T² ~ χ²(1). This relationship is useful in ANOVA and regression analysis where F-statistics are related to t-statistics.

Connection: t² with ν df equals χ² with 1 df

F-Distribution

The square of a t-random variable follows an F-distribution with (1, ν) degrees of freedom. This is fundamental in regression analysis and ANOVA.

Relationship: T²(ν) = F(1,ν)

Cauchy Distribution

When ν = 1, the t-distribution becomes the Cauchy distribution, which has undefined mean and variance due to its extremely heavy tails.

Special case: t(1) = Cauchy(0,1) - no finite moments

Advanced Topics

Welch's t-Test

When comparing two samples with unequal variances, Welch's t-test uses a modified degrees of freedom calculation (Welch-Satterthwaite equation) rather than the pooled variance approach.

Formula: df = (s₁²/n₁ + s₂²/n₂)² / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]

Multivariate t-Distribution

The multivariate extension is used in multivariate statistics, particularly in robust statistics and Bayesian analysis where it provides heavier tails than the multivariate normal distribution.

Application: Robust regression, outlier detection, and Bayesian inference

Non-Central t-Distribution

When the population mean is not zero under the alternative hypothesis, we get the non-central t-distribution, which is crucial for power analysis and sample size determination.

Use case: Power calculations for t-tests and determining required sample sizes

Conclusion

The t-distribution is an indispensable tool in statistical inference, particularly when dealing with small samples and unknown population parameters. Its robust theoretical foundation and practical applicability make it essential for hypothesis testing, confidence interval construction, and regression analysis.

Understanding when and how to apply the t-distribution, along with its assumptions and limitations, enables practitioners to make sound statistical decisions. Whether in medical research, quality control, or business analytics, the t-distribution provides a reliable framework for drawing inferences from limited data.

Key Takeaway: The t-distribution bridges the gap between theoretical statistics and practical application, providing robust inference methods when population parameters are unknown and sample sizes are small. Its heavier tails account for the additional uncertainty inherent in small-sample situations.