T-Distribution Table

What is the T-Distribution?

The t-distribution (Student's t) is a bell-shaped distribution used in place of the standard normal when the sample size is small (typically n < 30) or the population standard deviation σ is unknown. It is defined by its degrees of freedom (df = n − 1 for a one-sample test).

• As df → ∞, the t-distribution converges to the standard normal (Z).
• t-statistic: t = (x̄ − μ) / (s / √n), where s is the sample standard deviation.
• Two-tailed test: reject H₀ if |t| > t critical for the chosen α.

Critical T-Values (Two-Tailed)

Each cell shows the critical t-value t_{α/2, df} for a two-tailed test. The column headers show the total significance level α and the corresponding confidence level. For a one-tailed test, use the column where α equals twice your one-tailed significance level (e.g., for one-tailed α = 0.05, use the 0.10 column).

How to Use This Table

• Find df: for a one-sample t-test, df = n − 1. For a two-sample test, use the smaller of n₁ − 1 and n₂ − 1 (conservative) or Welch's formula.
• Choose α: select the column matching your significance level. For a 95% confidence interval, use the α = 0.05 column.
• Read the critical value: if |t calculated| > t critical, reject H₀.
• One-tailed test: use the column where the listed α equals twice your one-tailed α. For one-tailed α = 0.025, use the α = 0.05 column.
• Large samples: for df > 120, use the ∞ row — it equals the Z critical values.

T-Distribution vs. Z-Distribution

References

• Khan Academy — T-statistic and confidence intervals.
• OpenStax — Introductory Statistics, Section 8.2: Student's T-Distribution.
• Wikipedia — Student's t-distribution (definition, properties, and table).

See also