Direct definition
How to calculate critical value: choose the null distribution of the test statistic \(T\), pick the tail area(s) determined by \(\alpha\), then solve for the quantile cutoff \(c\) so that the tail probability equals \(\alpha\) (or \(\alpha/2\) in each tail for a two-tailed test).
Step 1: Identify the correct null distribution
Critical values come from the sampling distribution of the test statistic under \(H_0\). Common cases:
- z test: \(T=Z \sim N(0,1)\) under \(H_0\).
- t test: \(T \sim t_{\nu}\) under \(H_0\), with degrees of freedom \(\nu\).
- Chi-square: \(T \sim \chi^2_{\nu}\) under \(H_0\), often for variance tests or goodness-of-fit/independence.
- F test: \(T \sim F_{\nu_1,\nu_2}\) under \(H_0\), often for comparing variances or ANOVA.
Step 2: Determine whether the test is left-, right-, or two-tailed
The alternative hypothesis \(H_a\) fixes the rejection region:
- Right-tailed: \(H_a\) uses “greater than” (\(>\)); reject for large \(T\).
- Left-tailed: \(H_a\) uses “less than” (\(<\)); reject for small \(T\).
- Two-tailed: \(H_a\) uses “not equal” (\(\ne\)); reject for extreme values in both tails.
Step 3: Allocate the significance level \(\alpha\) to tail area(s)
The total probability of the rejection region under \(H_0\) must equal \(\alpha\):
- One-tailed: the single tail has area \(\alpha\).
- Two-tailed: each tail has area \(\alpha/2\).
Step 4: Use quantile equations to compute the critical value(s)
Let \(F_T\) be the CDF of the null distribution of \(T\). A critical value is a quantile:
- Left-tail cutoff \(c_L\) satisfies \[ P(T \le c_L)=\alpha \quad \Longleftrightarrow \quad c_L=F_T^{-1}(\alpha). \]
- Right-tail cutoff \(c_R\) satisfies \[ P(T \ge c_R)=\alpha \quad \Longleftrightarrow \quad P(T \le c_R)=1-\alpha \quad \Longleftrightarrow \quad c_R=F_T^{-1}(1-\alpha). \]
- Two-tailed cutoffs satisfy \[ c_L=F_T^{-1}\!\left(\frac{\alpha}{2}\right),\qquad c_R=F_T^{-1}\!\left(1-\frac{\alpha}{2}\right). \]
| Test type | Rejection region under \(H_0\) | Critical value equation(s) |
|---|---|---|
| Left-tailed | \(T \le c_L\) | \(c_L=F_T^{-1}(\alpha)\) |
| Right-tailed | \(T \ge c_R\) | \(c_R=F_T^{-1}(1-\alpha)\) |
| Two-tailed | \(T \le c_L\) or \(T \ge c_R\) | \(c_L=F_T^{-1}\!\left(\alpha/2\right)\), \(c_R=F_T^{-1}\!\left(1-\alpha/2\right)\) |
Common special forms for z and t critical values
When \(T\) is symmetric about \(0\) (standard normal \(Z\) and Student’s \(t\)), the two-tailed cutoffs are negatives of each other:
\[ c_L=-c_R. \]
For a z test:
- Right-tailed: \(c_R=z_{1-\alpha}\).
- Left-tailed: \(c_L=-z_{1-\alpha}\) (equivalently \(z_{\alpha}\)).
- Two-tailed: \(c_R=z_{1-\alpha/2}\) and \(c_L=-z_{1-\alpha/2}\).
For a t test, replace \(z_{\cdot}\) with \(t_{\cdot,\nu}\) using the same tail probabilities.
Visualization: critical values on a standard normal curve
Worked example (z critical value)
A population mean test uses a z statistic under \(H_0\). Suppose \(\alpha=0.05\) and the alternative is two-sided (\(\ne\)).
- Two-tailed implies \(\alpha/2=0.025\) in each tail.
- Right critical value is the \(1-\alpha/2=0.975\) quantile of \(N(0,1)\): \(c_R=z_{0.975}\approx 1.96\).
- Left critical value is the symmetric negative: \(c_L=-1.96\).
- Reject \(H_0\) when \(Z \le -1.96\) or \(Z \ge 1.96\).
Notes for \(\chi^2\) and F critical values
The chi-square and F distributions are not symmetric, so left and right critical values are not negatives of each other. The same quantile rules still apply:
- Right-tail \(\chi^2\) critical value: \(c_R=\chi^2_{1-\alpha,\nu}\) so that \(P(\chi^2_{\nu} \ge c_R)=\alpha\).
- Two-tail variance test: \(c_L=\chi^2_{\alpha/2,\nu}\) and \(c_R=\chi^2_{1-\alpha/2,\nu}\).
- Right-tail F critical value: \(c_R=F_{1-\alpha,\nu_1,\nu_2}\) so that \(P(F_{\nu_1,\nu_2} \ge c_R)=\alpha\).
Practical interpretation
A critical value is a cutoff chosen so that the probability of falling in the rejection region under \(H_0\) is exactly \(\alpha\). The calculation is therefore a quantile (inverse-CDF) step after selecting the correct distribution, degrees of freedom (if needed), and tail direction.