0.061146
0
1.25
0.061146
0
1.25
The T-Distribution Calculator computes the probability density function (PDF) and key moments of Student's t-distribution, one of the most important distributions in statistical inference. Developed by William Sealy Gosset (publishing under the pseudonym "Student" in 1908), the t-distribution arises whenever a population mean is estimated from a small sample when the population standard deviation is unknown. It is the foundation of the t-test family, one of the most widely used statistical procedures in science.
The t-distribution looks similar to the standard normal distribution — it is symmetric, bell-shaped, and centered at zero — but has heavier tails. These heavier tails account for the additional uncertainty introduced by estimating the population standard deviation from the sample. With few observations, the sample standard deviation is an imprecise estimate of the population value, so extreme values are more probable than under the normal distribution. As the degrees of freedom increase, the t-distribution converges to the standard normal, with the two being essentially identical for df > 30.
The degrees of freedom parameter ν controls the tail heaviness. With ν = 1, the t-distribution reduces to the Cauchy distribution, which has such heavy tails that its mean is technically undefined and its variance is infinite. For ν = 2, the variance is defined but infinite for practical purposes. For ν > 2, the variance is ν/(ν − 2), which approaches 1 as ν → ∞ (matching the standard normal variance). The t-distribution's heavier tails mean that confidence intervals and critical values are wider than their normal-theory counterparts, correctly reflecting the increased uncertainty from small samples.
In practice, the t-distribution is used for: one-sample and two-sample t-tests comparing means, paired t-tests for matched data, confidence intervals for means, and the coefficients in regression analysis (each regression coefficient's t-statistic follows a t-distribution under the null hypothesis). It is also used in Bayesian statistics as a robust alternative to the normal distribution for modeling data with potential outliers, where the degrees of freedom parameter controls the tail heaviness to accommodate occasional extreme observations.
The t-distribution belongs to the location-scale family, meaning that any linear transformation of a t-distributed variable is also t-distributed (with shifted location and scaled spread). It is also a special case of the generalized hyperbolic distribution. The kurtosis is 6/(ν − 4) for ν > 4, which exceeds the normal distribution's kurtosis of 0, quantifying the heavier tails. These mathematical properties underpin its central role in modern statistical inference and make it indispensable for researchers working with limited sample sizes.
The PDF of the t-distribution with ν degrees of freedom is:
$$f(t; \nu) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu + 1}{2}}$$
The distribution is symmetric about zero with moments:
$$\mu = 0 \;(\nu > 1), \quad \sigma^2 = \frac{\nu}{\nu - 2} \;(\nu > 2)$$
For computation, the PDF is evaluated in log-space using the Stirling approximation for the gamma function ratio, ensuring numerical stability for all degrees of freedom values.
The PDF value shows the relative likelihood of observing a particular t-value. Values near zero have the highest density, while extreme t-values (positive or negative) have low density — these correspond to the rejection regions of a t-test. The mean is always 0 (for ν > 1), reflecting the distribution's symmetry. The variance exceeds 1 for finite ν, reflecting the heavier tails compared to the standard normal. As df increases beyond 30, the variance approaches 1 and the distribution becomes nearly indistinguishable from the standard normal.
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At df=10, t=2.228 is the critical value for a two-tailed test at α=0.05. The low PDF (0.044) confirms this is in the tail region. The variance of 1.25 shows slightly heavier tails than the standard normal.
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Results
At df=100, the t-distribution closely approximates the standard normal. The variance of 1.02 is nearly 1, and the PDF at t=1.96 (0.059) is very close to the normal density of 0.058.
Use the t-distribution when estimating a population mean from a sample and the population standard deviation is unknown (which is almost always the case). For sample sizes below 30, the t-distribution's wider tails materially affect confidence intervals and p-values. For larger samples (n > 30), the t and normal distributions give nearly identical results, but using the t-distribution is always correct and never harmful. Most statistical software uses the t-distribution by default for mean inference.
For a one-sample t-test, df = n − 1. For a two-sample t-test with equal variances, df = n₁ + n₂ − 2. For Welch's t-test (unequal variances), df is calculated from the Satterthwaite approximation and may not be an integer. For paired t-tests, df = n − 1 where n is the number of pairs. In regression with p predictors and n observations, the residual df = n − p − 1.
The t-distribution accounts for the additional uncertainty from estimating the standard deviation. When you replace the true σ with the sample standard deviation s, you introduce extra variability. Sometimes s underestimates σ (making the t-statistic large) and sometimes it overestimates σ (making t small). This extra randomness spreads the distribution, creating heavier tails. With more data, s becomes a better estimate of σ, so the tails thin out and approach the normal distribution.
With df = 1, the t-distribution becomes the Cauchy distribution, which has no defined mean or variance. Its tails are so heavy that the sample mean does not converge to a fixed value as sample size increases — the law of large numbers fails. This makes the Cauchy distribution pathological for standard inference but useful in robust statistics (as a model for outlier-prone data) and in physics (it describes resonance line shapes in spectroscopy, known as the Lorentzian distribution).
Critical values are the t-values that cut off a specified tail probability. For a two-tailed test at α = 0.05 with df degrees of freedom, the critical values are ±t_{α/2, df}. Common values: df=10 → ±2.228, df=20 → ±2.086, df=30 → ±2.042, df=∞ → ±1.96 (normal). Reject H₀ if |t| exceeds the critical value. For one-tailed tests, use t_{α, df}. Statistical tables and software (including this calculator's PDF) help identify these thresholds.
Yes. The t-distribution is always symmetric about zero for all degrees of freedom. This symmetry follows from the definition: if Z is standard normal and V is chi-square with df degrees of freedom (independent of Z), then T = Z/√(V/df) has a t-distribution. Since Z is symmetric, so is T. This means P(T > c) = P(T < −c) for any value c, which simplifies hypothesis testing with two-tailed tests.
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