Student T Value Calculator
Calculate Student t critical values (t*) and p-values for one-tailed and two-tailed tests from degrees of freedom and significance level alpha.
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What Is a Student T Value Calculator?
The Student T Value Calculator computes critical values (t*) and p-values for Student’s t-distribution. It is used in one-sample, paired, and two-sample t-tests when the population standard deviation is unknown and sample size is small.
How to Use This Calculator
- Enter the degrees of freedom (df = n - 1 for one-sample tests).
- Choose your significance level alpha (α = 0.05, 0.01, 0.10).
- Select hypothesis tail direction (two-tailed or one-tailed).
- Optional: Enter a sample t-statistic to compute the exact p-value.
Student’s t-Distribution PDF
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}}Where ν is degrees of freedom (df) and Γ is the Gamma function. Critical values t* represent quantile points satisfying P(|T| > t*) = α.
Worked Example
Scenario: Finding the two-tailed critical t-value for df = 10 at α = 0.05
Degrees of freedom df = 10.
Two-tailed significance: α/2 = 0.025 in each tail.
Confidence level: 1 - 0.05 = 95%.
Evaluate quantile t*(0.975, 10) = 2.2281.
Tips & Key Notes
- For sample sizes n > 30, the critical t-value closely approaches the standard normal z-critical value (e.g., 1.96 for 95%).
- For unequal variance two-sample tests, calculate effective df using the Welch-Satterthwaite equation.
- Always use two-tailed critical values unless a directional hypothesis was specified prior to data collection.
Frequently Asked Questions
When should I use a t-test instead of a z-test?
Use a t-test when the population standard deviation (σ) is unknown and estimated using the sample standard deviation (s), especially with smaller sample sizes (n < 30).
What are degrees of freedom (df)?
Degrees of freedom represent the number of independent values that can vary in a statistical calculation. For a single sample mean, df = n - 1. For two independent samples with equal variance, df = n₁ + n₂ - 2.
What is the difference between one-tailed and two-tailed tests?
A two-tailed test evaluates whether a parameter is significantly different from the hypothesized mean, splitting α into both tails. A one-tailed test tests directional hypotheses (> or <) with the entire α in one tail.
What does the p-value mean in a t-test?
The p-value is the probability of observing a t-statistic as extreme as or more extreme than the calculated value, assuming the null hypothesis is true. If p < α, reject the null hypothesis.
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