Critical Value Calculator – Z, t, Chi-Square and F
The Critical Value Calculator on MyTimeCalculator brings together several key tools used in classical hypothesis testing and confidence interval construction. You can obtain Z, t, chi-square and F critical values for chosen significance levels, compute two-sided confidence interval multipliers and generate clean decision rules for rejecting or not rejecting a null hypothesis.
In many statistical procedures, the test statistic is compared to a critical value that depends on the chosen significance level \(\alpha\) and on the sampling distribution of the statistic under the null hypothesis. If the test statistic falls in a rejection region beyond the critical value, you conclude that the result is statistically significant at level \(\alpha\).
1. Z Critical Values
For a standard normal variable \(Z\), the critical values for a right-tailed, left-tailed or two-tailed test are derived from the standard normal cumulative distribution function \(\Phi(z)\).
- Right-tailed test: reject \(H_0\) if \(Z \ge z_{\text{crit}}\), where \[ P(Z \ge z_{\text{crit}}) = \alpha. \]
- Left-tailed test: reject \(H_0\) if \(Z \le z_{\text{crit}}\), where \[ P(Z \le z_{\text{crit}}) = \alpha. \]
- Two-tailed test: reject \(H_0\) if \(|Z| \ge z_{\text{crit}}\), where \[ P(|Z| \ge z_{\text{crit}}) = \alpha \quad\Rightarrow\quad z_{\text{crit}} = \Phi^{-1}(1 - \alpha/2). \]
2. t Critical Values
When the population standard deviation is unknown and sample sizes are not extremely large, test statistics often follow a Student t distribution with \(df\) degrees of freedom under the null hypothesis. Critical values for right-tailed, left-tailed and two-tailed t-tests are analogous to the Z case, but replace \(\Phi\) with the t distribution function \(F_t\) and \(\Phi^{-1}\) with the corresponding t quantile.
- Right-tailed: \(P(T \ge t_{\text{crit}} \mid df) = \alpha\).
- Left-tailed: \(P(T \le t_{\text{crit}} \mid df) = \alpha\).
- Two-tailed: \(P(|T| \ge t_{\text{crit}} \mid df) = \alpha\), so \(t_{\text{crit}} = t_{1-\alpha/2, df}\).
For very large \(df\), t critical values approach the corresponding Z critical values. For smaller sample sizes, they will be larger in magnitude, reflecting the extra uncertainty from estimating the standard deviation.
3. Chi-Square Critical Values
In many goodness-of-fit and independence tests, the test statistic follows a chi-square distribution under the null hypothesis. If \(X\) has a chi-square distribution with \(df\) degrees of freedom, the right-tail critical value for a given \(\alpha\) is defined by
The rejection region for a right-tailed chi-square test is \(\{X \ge \chi^2_{\text{crit}}\}\). The calculator uses a standard approximation to the chi-square distribution to provide practical values for many common degrees of freedom and significance levels.
4. F Critical Values
The F distribution arises as the ratio of two scaled chi-square variables and plays a central role in analysis of variance (ANOVA) and variance-ratio tests. If \(F\) has an F distribution with numerator degrees of freedom \(df_1\) and denominator degrees of freedom \(df_2\), then the right-tail critical value \(F_{\text{crit}}\) for significance level \(\alpha\) solves
In ANOVA, for example, you reject the null hypothesis of equal group means if the observed F statistic exceeds the critical value at the chosen \(\alpha\). The calculator provides approximate F critical values derived from standard relationships with chi-square distributions.
5. Confidence Interval Multipliers
Two-sided confidence intervals for means often have the form
For a known standard deviation and a normal model, the critical multiplier is \(z_{1-\alpha/2}\). When the standard deviation is estimated and the sample size is moderate, the multiplier becomes \(t_{1-\alpha/2,df}\). The confidence interval tab in this calculator returns the relevant critical multiplier given a confidence level and, where needed, degrees of freedom.
6. Decision Rules for Hypothesis Tests
A decision rule summarizes how the test statistic will be compared to critical values. For example, in a right-tailed Z-test at significance level \(\alpha = 0.05\), a typical rule is:
For a two-tailed t-test with \(df\) degrees of freedom, the rule becomes:
The decision rule generator tab applies this logic to Z, t, chi-square and F distributions, automatically producing a readable rule based on your selected distribution, degrees of freedom, tail type and significance level.
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Critical Value Calculator FAQs
Frequently Asked Questions
Quick answers to common questions about Z, t, chi-square and F critical values, confidence intervals and hypothesis test decision rules.
A critical value is a cutoff point on the sampling distribution of a test statistic. Given a significance level \(\alpha\) and a null hypothesis, it defines a rejection region: if the observed test statistic falls beyond the critical value in that region, the result is declared statistically significant at level \(\alpha\). Different tests use different distributions and, therefore, different critical values.
Z critical values are used when the population standard deviation is known and the sampling distribution of the statistic is effectively normal, or when sample sizes are very large so that the normal approximation is accurate. t critical values are used when the standard deviation is estimated from the data and sample sizes are more modest, in which case the test statistic follows a t distribution with finite degrees of freedom under the null hypothesis.
In a one-tailed test, the rejection region is entirely in one tail of the distribution (left or right), so the full significance level \(\alpha\) is allocated to that tail. In a two-tailed test, the rejection region is split between the two tails, with \(\alpha/2\) in each tail. As a result, two-tailed critical values are larger in magnitude than the corresponding one-tailed critical values for the same \(\alpha\).
Two-sided confidence intervals at confidence level \(1 - \alpha\) use the same critical values as two-tailed tests at significance level \(\alpha\). For example, a 95% confidence interval for a mean often uses \(z_{0.975}\) or \(t_{0.975, df}\) as the multiplier. The confidence interval multiplier tab in this calculator returns exactly these values for your chosen confidence level and degrees of freedom.
Z critical values are based on the standard normal distribution and are very accurate. t and chi-square critical values are computed using well-known approximations that are typically precise enough for many applied and educational uses. F critical values are derived using approximate relationships with chi-square distributions. For high-stakes or highly sensitive analyses, you may wish to verify values using specialized statistical software or detailed distribution tables.