Student's t-Distribution Calculator

Calculate t-distribution probabilities, critical values, and confidence intervals. Essential for hypothesis testing and statistical analysis.

Probability (CDF)
Critical Value
Confidence Interval

Calculate Probability: Find P(T ≤ t) or P(|T| ≥ |t|) for a given t-value and degrees of freedom

Must be a positive integer (typically n-1 for sample size n)
The t-statistic value
Left-tailed: P(T ≤ t)
Right-tailed: P(T ≥ t)
Two-tailed: P(|T| ≥ |t|)
Calculating...

Understanding Student's t-Distribution

The Student's t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and/or when the population variance is unknown. It was first published by William Sealy Gosset under the pseudonym "Student" in 1908.

Key Properties:

  • Bell-shaped and symmetric like the normal distribution
  • Heavier tails than the normal distribution (more probability in the tails)
  • Shape depends on degrees of freedom (df)
  • As df → ∞, t-distribution approaches the standard normal distribution
  • Used in t-tests, confidence intervals, and linear regression

Probability Density Function (PDF)

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

Where:

  • t is the t-statistic
  • ν (nu) is the degrees of freedom
  • Γ is the gamma function
  • π is pi (approximately 3.14159)


When to Use t-Distribution

1

Small Sample Sizes: When sample size n < 30, the t-distribution provides more accurate results than the normal distribution.

2

Unknown Population Variance: When the population standard deviation is unknown and must be estimated from the sample.

3

Normality Assumption: The underlying population should be approximately normally distributed, though the t-test is robust to mild violations of this assumption.

Common Applications

  • One-sample t-test: Testing if a sample mean differs from a known population mean
  • Two-sample t-test: Comparing means of two independent samples
  • Paired t-test: Comparing means of paired or matched samples
  • Confidence intervals: Estimating population mean with a margin of error
  • Regression analysis: Testing significance of regression coefficients

Critical t-Values Table (Two-tailed, α = 0.05)

df t-critical df t-critical df t-critical
1 12.706 11 2.201 30 2.042
2 4.303 12 2.179 40 2.021
3 3.182 13 2.160 50 2.009
4 2.776 14 2.145 60 2.000
5 2.571 15 2.131 80 1.990
6 2.447 16 2.120 100 1.984
7 2.365 17 2.110 120 1.980
8 2.306 18 2.101 ∞ (normal) 1.960
9 2.262 19 2.093
10 2.228 20 2.086

Frequently Asked Questions

Degrees of freedom (df) represent the number of independent values in a calculation that can vary. For a t-test, df = n - 1, where n is the sample size. The degrees of freedom affect the shape of the t-distribution - with fewer df, the distribution has heavier tails.

Use a one-tailed test when you have a directional hypothesis (e.g., "Treatment A is greater than Treatment B"). Use a two-tailed test when you have a non-directional hypothesis (e.g., "Treatment A is different from Treatment B"). Two-tailed tests are more conservative and are generally preferred unless you have strong theoretical reasons for a directional hypothesis.

The t-distribution is similar to the normal distribution but has heavier tails. This means extreme values are more likely in a t-distribution. As the degrees of freedom increase, the t-distribution approaches the normal distribution. For df ≥ 30, the difference is minimal for most practical purposes.

The confidence level and significance level are complementary. A 95% confidence level corresponds to a 5% significance level (α = 0.05). In other words, if you're 95% confident that your interval contains the true parameter, there's a 5% chance it doesn't (Type I error rate).

Yes, you can use the t-distribution for any sample size, but as the sample size increases (typically n > 30), the t-distribution converges to the normal distribution. For large samples, the difference between using t-distribution and normal distribution is negligible, but using the t-distribution is always theoretically correct when the population variance is unknown.