Critical Value Calculator

A critical value is the cutoff point on a test statistic's distribution that separates the region where you fail to reject the null hypothesis from the rejection region. It is determined entirely by the chosen distribution, the significance level (α), and the tail of the test — not by your data.

• Rejection region: If your test statistic falls beyond the critical value, you reject the null hypothesis.
• One-tailed vs. two-tailed: Two-tailed tests split α across both tails, producing a more extreme cutoff than a one-tailed test at the same α.
• Inverse of the p-value: A critical value answers "how extreme must my statistic be?", while a p-value answers "how extreme is the statistic I already have?".

Traditional critical-value tables are indexed by the significance level (and, where relevant, the degrees of freedom). To use one, find the column for your α and tail, then read across the row for your degrees of freedom:

• Z (standard normal): Two-tailed at α = 0.05 gives $\pm 1.96$; right-tailed gives $1.645$.
• t-distribution: Requires df. As df grows, the t critical value approaches the z critical value.
• Chi-square & F: Asymmetric and non-negative. Two-tailed tests return a lower and upper bound, commonly used for confidence intervals on a variance.
• Pearson r: The critical correlation is derived from the t-distribution with $df = n - 2$: $r_{crit} = t / \sqrt{df + t^2}$.

Critical values and p-values are two sides of the same decision. They always agree:

• Critical-value approach: Reject $H_0$ if your test statistic is more extreme than the critical value.
• P-value approach: Reject $H_0$ if the p-value is less than α.

For example, a two-tailed z-test at α = 0.05 has a critical value of $\pm 1.96$. A test statistic of $z = 2.1$ exceeds 1.96, so you reject $H_0$ — equivalently, its p-value (0.0357) is below 0.05. To go the other direction and compute a p-value from a statistic, use the P-Value Calculator.

• Z (standard normal): Large samples or a known population standard deviation; proportions.
• t (Student's t): Small samples with an unknown population standard deviation; means and regression coefficients.
• Chi-square (χ²): Goodness-of-fit, tests of independence, and variance.
• F: Comparing variances and ANOVA.
• Pearson r: Testing whether a sample correlation is significantly different from zero.

In R, use the quantile functions qnorm() for Z, qt() for t, qchisq() for Chi-Square, and qf() for F. The leading 'q' stands for quantile (the inverse CDF). Here are some examples:

alpha <- 0.05

# Z critical value (two-tailed): +/- this value
qnorm(1 - alpha/2)            # 1.959964

# t critical value (right-tailed), df = 10
qt(1 - alpha, df = 10)        # 1.812461

# Chi-square critical value (right-tailed), df = 5
qchisq(1 - alpha, df = 5)     # 11.0705

# F critical value (right-tailed), df1 = 3, df2 = 20
qf(1 - alpha, df1 = 3, df2 = 20)  # 3.098391

# Critical Pearson r (two-tailed), n = 20 -> df = 18
df <- 20 - 2
t_crit <- qt(1 - alpha/2, df)
t_crit / sqrt(df + t_crit^2)  # 0.4438

In Python, use the percent-point function .ppf() from scipy.stats: stats.norm for Z, stats.t for t, stats.chi2 for Chi-Square, and stats.f for F. Here are some examples:

from scipy import stats
import numpy as np

alpha = 0.05

# Z critical value (two-tailed): +/- this value
stats.norm.ppf(1 - alpha/2)          # 1.959964

# t critical value (right-tailed), df = 10
stats.t.ppf(1 - alpha, df=10)        # 1.812461

# Chi-square critical value (right-tailed), df = 5
stats.chi2.ppf(1 - alpha, df=5)      # 11.0705

# F critical value (right-tailed), df1 = 3, df2 = 20
stats.f.ppf(1 - alpha, dfn=3, dfd=20)  # 3.098391

# Critical Pearson r (two-tailed), n = 20 -> df = 18
df = 20 - 2
t_crit = stats.t.ppf(1 - alpha/2, df)
t_crit / np.sqrt(df + t_crit**2)     # 0.4438