T Distribution Calculator
Calculate Student's $t$-distribution cumulative probabilities, one-tailed and two-tailed $p$-values, and degrees of freedom ($df$).
Calculate Student's t-Distribution p-Values
Enter t-statistic value, degrees of freedom (df), and tail test type.
Calculation Results
Calculated using Student's t-distribution PDF integration: 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}}
Quick Summary
The T Distribution Calculator evaluates Student's $t$-distribution probabilities, two-tailed $p$-values ($P(|T| > |t|)$), one-tailed $p$-values, and significance ($p < 0.05$).
Formula Explanation
\text{Two-Tailed p-value} = 2 \times P(T > |t|)
\text{One-Tailed p-value} = P(T > t)How It Works
Student's $t$-distribution models small sample statistics with $df$ degrees of freedom. The T Distribution Calculator evaluates exact $p$-values for t-tests, determining statistical significance without relying on printed $t$-tables.
Step-by-Step Worked Example
Practical Problem: Find the two-tailed $p$-value for calculated $t = 2.228$ with $df = 10$ degrees of freedom.
- Step 1: Identify t-statistic and degrees of freedom: $t = 2.228$, $df = 10$.
- Step 2: Evaluate upper one-tail probability $P(T > 2.228)$: $P(T > 2.228) = \mathbf{0.0250}$.
- Step 3: Calculate two-tailed p-value ($2 \times 0.0250$): $p = \mathbf{0.0500}$.
- Step 4: Compare p-value to alpha threshold ($\alpha = 0.05$): $p = 0.0500 \le 0.05$.
- Step 5: Interpretation: The result is statistically significant at $\alpha = 0.05$ significance level.
Real-World Calculation Examples
Scenario 1: t = 2.228, df = 10 (Two-Tailed)
Parameters: t = 2.228, df = 10
Result: p = 0.0500 (Significant at 5% level).
Scenario 2: t = 3.169, df = 10 (High Significance)
Parameters: t = 3.169, df = 10
Result: p = 0.0100 (Significant at 1% level).
Scenario 3: Small Sample Trial (df = 4)
Parameters: t = 2.132, df = 4
Result: p = 0.1000 (Not significant at 5%).
Scenario 4: Large Sample Convergence (df = 100)
Parameters: t = 1.984, df = 100
Result: p = 0.0500 (Approaches Z = 1.96).
Key Benefits of Using This Calculator
Instant p-Value Integration
Calculates exact one-tailed and two-tailed $p$-values for any $df$.
Automated Significance Flag
Automatically flags results as statistically significant ($p < 0.05$).
Replaces Paper T-Tables
Provides exact probabilities instead of coarse paper table ranges.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is Student's t-distribution?
Student's $t$-distribution is a bell-shaped probability distribution symmetric around 0, used when sample size is small ($n < 30$) and population standard deviation is unknown.
How does degrees of freedom (df) affect the t-distribution curve?
Lower $df$ creates heavier tails and a lower peak; as $df \rightarrow \infty$, the t-distribution becomes identical to standard normal distribution $N(0,1)$.
What is difference between one-tailed and two-tailed t-test?
A one-tailed test tests directional effect (greater or less); a two-tailed test tests any directional difference (unequal).
How do I calculate t-distribution p-values in Excel?
Use formula =T.DIST.2T(t, df) for two-tailed or =T.DIST.RT(t, df) for right one-tailed.
What is critical t-value for 95% confidence with df = 10?
For two-tailed $\alpha = 0.05$ with $df = 10$, critical $t = 2.228$.
What is relationship between t-distribution and F-distribution?
The square of a $t$-distributed random variable with $df$ degrees of freedom is $F$-distributed with $1$ and $df$ degrees of freedom ($t^2 = F_{1, df}$).
Can t-statistic be negative?
Yes — $t$ is symmetric around 0, so $P(T < -t) = P(T > t)$.
What is null hypothesis in t-test?
Null hypothesis ($H_0$) assumes no significant difference between means ($\mu_1 - \mu_2 = 0$).
What is alpha (α) level?
Alpha ($\alpha$, typically $0.05$) is the probability threshold for rejecting the null hypothesis (Type I error rate).
When should I use Z-test instead of t-test?
Use Z-test when population standard deviation $\sigma$ is known or when sample size is very large ($n > 100$).