https://doi.org/10.1351/goldbook.C01247

Symmetric confidence limits (\(\pm C\)) about the estimated mean, which cover the population mean with @[email protected] \(1 - \alpha \). The quantity \(C\) is calculated by the formula: \[C = \frac{t_{\text{p},v^{S}}}{\sqrt{n}}\] Here \(t_{\text{p},v}\), is the critical value from the \(t\)- (or Student) distribution function corresponding to the @[email protected] \(1 - \alpha \) and @[email protected] \(v\). The symbol \(p\) represents the percentile (or percentage point) of the \(t\)-distribution. For 1-sided intervals, \(p=1- \alpha \); for 2-sided intervals, \(p=1- \frac{\alpha }{2}\). In each case, the @[email protected] is \(1 - \alpha \). The confidence interval is given as \(\overline{x}\ \pm \ C\).

If the population @[email protected] \(σ\) is known, confidence limits about a single result may be calculated with the formula: \[C = t_{\text{p},\infty}\sigma\] The @[email protected] \(t_{p,\infty}\), is the limiting value of the \(t\)-distribution function for \(\nu = \infty\) at @[email protected] \(1 - \alpha\). This is identical to \(z_{\text{p}}\), the \(p\)th percentage point of the standard normal variate.

**Note:**

If the population @[email protected] \(σ\) is known, confidence limits about a single result may be calculated with the formula: \[C = t_{\text{p},\infty}\sigma\] The @[email protected] \(t_{p,\infty}\), is the limiting value of the \(t\)-distribution function for \(\nu = \infty\) at @[email protected] \(1 - \alpha\). This is identical to \(z_{\text{p}}\), the \(p\)th percentage point of the standard normal variate.