T Distribution Calculator

Compute probabilities (p-values), critical t-values, and visualize the t-curve. Essential for t-tests, confidence intervals, and regression analysis. Interactive graph with shading for one-tail or two-tail regions.

Find Probability (p-value)
Tail type:
P(T ≤ t) = —
P(T > t) = —
Two-tailed p-value = —
Find Critical t-value (Quantile)
Critical t (two-tailed, α/2) = —
For one-tailed test, use α (two-tailed α/2 corresponds to one-tailed α/2).
? Classic: t=2.0, df=20 (two-tailed p≈0.059)
? t=1.645, df=100 (z approx, one-tail 0.05)
? t=2.776, df=4 (critical 95% two-tail)
? t=1.812, df=10 (one-tail 0.05)
✏️ α=0.01, df=25 (critical ≈2.787)
Privacy first: All calculations use jStat (client-side). No data leaves your browser. Graphics rendered locally.
Student's t-Distribution Curve
PDF (t-distribution)
Rejection region (shaded)
t = —

Understanding the t-Distribution

The Student's t-distribution (or simply t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym "Student". The t-distribution is symmetric and bell-shaped, but has heavier tails compared to the normal distribution, especially for small degrees of freedom (df). As df increases, the t-distribution approaches the standard normal distribution (Z).

Probability Density Function (PDF):

\( f(t) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)} \left(1+\frac{t^{2}}{\nu}\right)^{-\frac{\nu+1}{2}} \)

where ν = degrees of freedom, Γ is the gamma function.

Key Applications & Real-World Use

  • One-sample t-test: Compare sample mean to a known population mean.
  • Two-sample t-test: Compare means from two independent groups.
  • Paired t-test: Before-after studies.
  • Confidence Intervals: Estimate population mean from small samples.
  • Linear Regression: Test significance of coefficients (t-statistic).

How to Use This Calculator

For probability (p-value): Enter a t-score and degrees of freedom, choose tail type (two-tailed, right-tailed, left-tailed). The calculator returns the cumulative probability P(T ≤ t), the survival P(T > t), and the two-tailed p-value. The graph dynamically shades the region corresponding to the selected tail(s) based on your t-value.

For critical value (inverse CDF): Provide significance level α (commonly 0.05 for 95% confidence) and degrees of freedom. The tool returns the two-tailed critical t-value, i.e., the value such that P(|T| > t_crit) = α. This value is essential for hypothesis testing thresholds.

Step-by-Step Example: Two-Tailed t-Test

Clinical Trial: Drug Efficacy

A researcher tests a new drug on 25 patients (df=24). The computed t-statistic from the study is 2.064. Using our calculator with df=24, t=2.064 (two-tailed), we obtain a p-value ≈ 0.05. Since p ≤ 0.05, the result is statistically significant at the 5% level, suggesting the drug has a measurable effect. The shaded region on the curve illustrates the extreme 5% area beyond ±2.064, supporting the decision to reject the null hypothesis.

Properties of t-Distribution

Property Description
Mean 0 (for ν > 1)
Variance ν/(ν−2) for ν > 2; infinite for ν ≤ 2
Skewness 0 (symmetric)
Kurtosis 3 + 6/(ν−4) for ν > 4
Support t ∈ (−∞, +∞)

Frequently Asked Questions

Use t-distribution when the population standard deviation is unknown and sample size is small (typically n < 30). For large samples, the t-distribution approximates the normal distribution closely.

Degrees of freedom (df) represent the number of independent pieces of information available. For one-sample t-test, df = n-1. Higher df yields a curve closer to the standard normal.

Yes, the t-distribution is symmetric about 0. Negative t-values indicate the sample mean is below the hypothesized value; probabilities are computed using the same distribution.

We use jStat, a well-established JavaScript statistical library with high numerical precision (double-precision). Results are accurate to at least 6 decimal places, suitable for scientific research and coursework.

The t-test statistic follows a t-distribution under the null hypothesis. Our calculator converts t-values to p-values, enabling decision-making in hypothesis testing.

Rooted in statistical excellence – This tool implements the Student's t-distribution based on the mathematical formulations by William Gosset and later refinements by R.A. Fisher. Calculations are validated against standard statistical tables (e.g., Pearson & Hartley, 1970). The visual engine renders the PDF accurately via numerical evaluation. Reviewed by the GetZenQuery tech team, last update May 2026.

References: MathWorld: Student's t-Distribution; Casella, G., & Berger, R. L. (2002). Statistical Inference; Wikipedia: t-distribution.