A critical value, in the construction of a [[confidence interval]] or [[hypothesis testing]], is the number that cuts off a specified area under a [[probability density function]], often denoted as $\alpha$. $\alpha$ equals $1$ minus the confidence level (e.g., for a $95\%$ confidence level, $\alpha = 0.05$). In statistics and math in general, a value denoted by $\alpha$ is expected to be small. We follow the standard to denote **area $\alpha$ to the right** with the critical value; however other references may use the opposite standard. Keep in mind that when calculating critical values in [[R]], the area to the left ($1 - \alpha$) should be provided. For example, to get the critical value for the standard normal distribution $z_{\alpha}$: ```R qnorm(1 - alpha) ``` ## Critical values for common distributions | Distribution | Notation | Degrees of Freedom | | -------------------------------- | ---------------------------- | ------------------ | | [[standard normal distribution]] | $z_\alpha$ | N/A | | [[t-distribution]] | $t_{\alpha, \ df}$ | $n - 1$ | | [[chi-squared distribution]] | $\chi^2_{\alpha, \ df}$ | $n - 1$ | | [[F-distribution]] | $F_(\alpha, \ df1, \ df2)$ | $n_1 - 1, n_2 - 1$ |