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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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.
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.
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.
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).
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.
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.
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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