Home / Shared Math Details / Distribution Functions / Student t Distribution
Student t Distribution¶
This page defines the Student t formulas Quantum XL shares across tools: the two-sided p-value for a test statistic, and the one-sample confidence interval for the mean.
Notation¶
| Term | Description |
|---|---|
| \(T_{df}\) | Student t cumulative distribution function with \(df\) degrees of freedom |
| \(t\) | a test statistic |
| \(df\) | degrees of freedom |
| \(\alpha\) | significance level (\(1 -\) confidence) |
| \(\bar{x}\) | sample mean |
| \(s\) | sample standard deviation (n − 1 denominator) |
| \(n\) | sample size (sum of weights when frequency-weighted) |
Two-sided p-value¶
For a test statistic \(t\) with \(df\) degrees of freedom, the two-sided p-value is:
\[ p = 2\left(1 - T_{df}\!\left(|t|\right)\right) \]
Used by:
- Correlation and Covariance: the correlation significance test, with \(df = n - 2\) and \(t = r\sqrt{df / (1 - r^2)}\).
- Scatter Plot: the regression coefficient t-tests, with \(df = n - p\) and \(t_j = \hat{\beta}_j / \mathrm{SE}(\hat{\beta}_j)\).
- Time Series: the autocorrelation significance limits.
One-sample confidence interval for the mean¶
\[ \bar{x} \pm t_{\,df,\,1-\alpha/2}\,\frac{s}{\sqrt{n}}, \qquad df = n - 1 \]
where \(t_{\,df,\,1-\alpha/2}\) is the \((1 - \alpha/2)\) quantile of the Student t distribution.
Used by: Confidence Interval and Summary Statistics.
See Also¶
References¶
- Casella, G., and Berger, R. L. (2002). Statistical Inference, 2nd ed. Duxbury.
- NIST/SEMATECH e-Handbook of Statistical Methods, section on the t distribution.