t
Your calculated t-value
df
n − 1 for one sample; n₁+n₂−2 for two samples
P-Value
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Understanding the T-Distribution
The Student's t-distribution is used when analyzing sample means, especially for small samples (n < 30) or when the population standard deviation is unknown. It was developed by William Gosset in 1908 under the pseudonym "Student."
t = (x̄ − μ₀) ÷ (s ÷ √n)
x̄ = sample mean, μ₀ = hypothesized mean, s = sample std dev, n = sample size
Degrees of freedom = n − 1 for one-sample t-test
Degrees of freedom = n − 1 for one-sample t-test
Common T-Test Types
- One-sample t-test: Is the sample mean different from a known value? df = n−1
- Two-sample t-test: Are two independent group means different? df = n₁+n₂−2
- Paired t-test: Are two related measurements different (before/after)? df = n−1 (n = number of pairs)
💡 As df → ∞, the t-distribution converges to the standard normal distribution (Z). For df > 30, results are very close to Z-score results.
Critical T-Value Reference Table
Common critical t-values for two-tailed tests at α = 0.05:
- df = 5: ±2.571 | df = 10: ±2.228 | df = 15: ±2.131
- df = 20: ±2.086 | df = 30: ±2.042 | df = 60: ±2.000
- df = 120: ±1.980 | df = ∞ (z): ±1.960
At α = 0.01 (two-tailed): df=10 → ±3.169; df=20 → ±2.845; df=30 → ±2.750. As sample size grows, t-values approach the z-value of 1.96.
Frequently Asked Questions
Use a t-test when the population standard deviation is unknown and you are estimating it from the sample (most real-world cases). Use a z-test when the population standard deviation is known OR when sample size is very large (n > 100). In practice, t-tests are far more common because population standard deviations are almost never known in advance.
A two-tailed test checks whether the mean differs in either direction from the null hypothesis (H₁: μ ≠ μ₀). A one-tailed test checks whether it differs in one specific direction (H₁: μ > μ₀ or H₁: μ < μ₀). Two-tailed tests are more conservative and are the default recommendation. Use one-tailed tests only when you have a strong directional hypothesis specified before collecting data.
Critical t-values (two-tailed, α=0.05): df=10 → ±2.228; df=20 → ±2.086; df=30 → ±2.042; df=60 → ±2.000; df=120 → ±1.980; df=∞ (z) → ±1.960. As sample size increases, the critical t-value approaches 1.96 (the z-value). For small samples (df<10), the critical value is substantially higher, requiring stronger evidence for significance.
The p-value is the probability of observing a t-statistic as extreme as yours (or more extreme) if the null hypothesis were true. A p-value of 0.03 means: if the null hypothesis were correct, there would be a 3% chance of seeing results this extreme by random chance alone. This is below the conventional 0.05 threshold, so you reject the null hypothesis.
One-sample t-test: df = n − 1. Two-sample independent t-test: df = n₁ + n₂ − 2 (equal variances) or Welch's formula for unequal variances. Paired t-test: df = n − 1 where n is the number of pairs. Chi-square test: df = categories − 1 for goodness-of-fit, or (rows−1)×(cols−1) for contingency tables. Degrees of freedom reflect how many independent pieces of information went into estimating a parameter.
Use a power analysis to determine the required sample size. For a typical two-sample t-test with medium effect size (Cohen's d = 0.5), α = 0.05, and 80% power, you need about 64 subjects per group (128 total). For a large effect (d = 0.8), you need about 26 per group. Small samples can still yield valid results, but require larger effects to achieve significance. Many researchers use G*Power (free software) for these calculations.
Sources & Methodology
This calculator's logic and benchmarks are based on the following authoritative sources, reviewed regularly for accuracy.
NIST/SEMATECH Statistical Methods Handbook
T-distribution and t-test methodology — nist.gov
Khan Academy — Significance Tests
T-test instruction and worked examples — khanacademy.org
Gosset, W.S. (1908) "The Probable Error of a Mean"
Original Student t-distribution paper — Biometrika
Methodology: T-Distribution P-Value Calculations calculations use industry-standard formulas verified against the sources above. Results are estimates — consult a qualified professional for significant financial, legal, or structural decisions.
Last reviewed: March 2026
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