Visual Tool
BY: Statistics Fundamentals Team  |  Reviewed By: Minsa A (Senior Statistics Editor)

t-Distribution Visualizer

Explore how the Student’s t-distribution changes shape with degrees of freedom. Drag the slider to watch the distribution converge toward the standard normal. Find exact critical values, compute p-values from any test statistic, and see how CI width narrows as df grows.

t-Distribution Visualizer

PDFf(t) ∝ (1 + t²/ν)^(−(ν+1)/2) ν→∞t → N(0,1)
10
Quick df:
ν (df)10
Variance1.25
t* (95% CI)2.228
vs z*=1.960+13.7%

t(10) distribution

Distribution Properties

Degrees of freedom10
Mean0
Variance ν/(ν−2)1.250
Std deviation1.118
Excess kurtosis1.000
Skewness0 (symmetric)

Critical Values (this df)

t* for 90% CI—
t* for 95% CI—
t* for 99% CI—
t* one-tail α=0.05—
t* one-tail α=0.01—
z* for reference (∞)z*=1.960 (95%)

How t* Converges to z* (95% CI)

Bar length = difference between t* and z*=1.960. Shorter bar = closer to normal.

Critical Value Table (t* vs z*)

dfα=0.10 (90%)α=0.05 (95%)α=0.02 (98%)α=0.01 (99%)α=0.001 (99.9%)
p-valueP(|T| ≥ |t_obs| | H₀) = area in tails

Enter your observed t-statistic and degrees of freedom to get the exact p-value with a shaded diagram showing the tail probability.

Quick examples:

CI formulax̄ ± t* · s/√n  →  t* shrinks from 12.706 (df=1) to 1.960 (∞)

The table below shows how the 95% CI half-width multiplier (t*) shrinks as degrees of freedom increase. Enter a standard deviation and sample size to see actual CI widths for your data.

t* Convergence to z*=1.960 (95% CI)

CI Half-Width (ME) by df and n

Degrees of freedomdf = n − 1 for one-sample t-test

Each example below represents a real study design. Click any card to load the exact df, see the t-distribution shape, and get the critical value that applies.

What Is the t-Distribution?

The Student’s t-distribution is a family of symmetric, bell-shaped distributions parameterized by degrees of freedom (ν). Developed by William Sealy Gosset in 1908 (published as “Student”), it solves the problem of estimating a population mean when σ is unknown and n is small. The t-distribution has heavier tails than the normal, making critical values larger and confidence intervals wider — an honest accounting of extra uncertainty from estimating σ with s.

Degrees of Freedom and When t Converges to Normal

As ν increases, the t-distribution converges to N(0,1). The variance of t(ν) = ν/(ν−2) approaches 1 from above. By ν=30, the 95% critical value t*=2.042 differs from z*=1.960 by only 4.2%. By ν=120, the difference is under 1%. In practice, researchers use z-intervals when n ≥ 30 as a pragmatic approximation, but the t-interval is always correct when σ is unknown.

How the t* Critical Value Shrinks

df (n−1)t* (95% CI)z* (reference)DifferenceExample study
112.7061.960+548%n=2, pilot study
42.7761.960+42%n=5, small lab group
92.2621.960+15%n=10, small clinical trial
152.1311.960+8.7%n=16, blood pressure study
192.0931.960+6.8%n=20, psychology experiment
292.0451.960+4.3%n=30, class survey
592.0011.960+2.1%n=60, field study
1191.9801.960+1.0%n=120, large survey
∞1.9601.9600%population or very large n

Degrees of Freedom for Different Tests

One-sample t-test: df = n − 1. One df is “used up” estimating μ with x̄.
Two-sample independent t-test (equal variances): df = n₁ + n₂ − 2.
Paired t-test: df = n − 1, where n = number of pairs.
Simple linear regression (slope): df = n − 2 (one df each for slope and intercept).
Welch’s t-test (unequal variances): df estimated by the Welch-Satterthwaite equation, often non-integer.

Related Topics

t-Distribution Guide T-Test Calculator CI Visualizer p-Value Visualizer Normal Distribution Guide Z-Score to Percentile

Sources:

  • Student [Gosset, W.S.] (1908). The Probable Error of a Mean. Biometrika, 6(1), 1–25.
  • NIST Engineering Statistics Handbook — t-Distribution
  • Abramowitz, M. & Stegun, I.A. (1965). Handbook of Mathematical Functions. Dover Publications.

Frequently Asked Questions

The t-distribution accounts for two sources of uncertainty: variability in the data AND uncertainty in estimating σ from s. When n is small, this extra uncertainty inflates the tails, requiring larger critical values to achieve the same α level. This makes inferences more honest — we pay for not knowing σ with wider intervals.

By df=30, t*(95%) = 2.042 vs z*=1.960, a difference of 4.2%. By df=120, the gap is under 1%. The common rule “use z when n ≥ 30” is a pragmatic approximation; technically you should always use t when σ is estimated from data. For df ≥ 100, the choice makes virtually no practical difference.

Var(T) = ν/(ν−2) for ν > 2. This is always > 1 (the normal’s variance), reflecting heavier tails. For df=3, Var=3; df=5, Var=5/3=1.667; df=10, Var=10/8=1.25; df=30, Var=30/28≈1.07. As ν→∞, Var→1. For ν≤2, variance is undefined (infinite), and for ν=1 (Cauchy), the mean is also undefined.

We estimate the population mean μ using the sample mean x̄. Because the deviations (xᵢ−x̄) must sum to zero, once you know n−1 of them, the last is determined — there are only n−1 independent pieces of information. This “used up” df is what makes the sample variance s² an unbiased estimator: dividing by n−1 instead of n corrects for the constraint imposed by using x̄.

For a two-tailed test: p = 2 × P(T ≥ |t₀bs| | H₀) = 2 × (1 − CDF(|t₀bs|, df)). Use the p-Value Calculator tab above to compute this exactly. The CDF of the t-distribution involves the regularized incomplete beta function: CDF(t, ν) = I(ν/(ν+t²); ν/2; 1/2) for t < 0.