{
  "id": "critical-value",
  "version": "4de8f84bf28f",
  "status": "published",
  "name": "Critical Value Calculator",
  "question": "What is the critical value for my test?",
  "summary": "Computes the critical values and rejection region of a left-, right-, or two-tailed test for the z, t, chi-square, and F distributions at a significance level.",
  "category": "statistics",
  "subcategory": "tests",
  "url": "https://www.acalculator.org/statistics/critical-value-calculator",
  "markdown": "https://www.acalculator.org/statistics/critical-value-calculator.md",
  "kind": "function",
  "method": "Two-tailed: the α/2 and 1 − α/2 quantiles; left-tailed: the α quantile; right-tailed: the 1 − α quantile of the chosen distribution.",
  "assumptions": [
    "The critical values invert the CDF numerically, not read from a table. They are good to about 10 significant digits or better (13 for most inputs).",
    "Degrees of freedom may be fractional (for example Welch’s t-test), from 1 up to 100,000.",
    "The rejection region excludes its boundary, and shows each critical value to 4 decimal places, at most 8 significant figures."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "distribution": {
        "title": "Distribution",
        "description": "The distribution of the test statistic: z (standard normal), Student’s t, chi-square, or F.",
        "type": "string",
        "enum": [
          "z",
          "t",
          "chi-square",
          "f"
        ]
      },
      "testType": {
        "title": "Test type",
        "description": "Where the rejection region lies: both tails, the left tail, or the right tail.",
        "type": "string",
        "enum": [
          "two-tailed",
          "left-tailed",
          "right-tailed"
        ]
      },
      "significanceLevel": {
        "title": "Significance level (α)",
        "description": "The probability of the rejection region when the null hypothesis is true, for example 0.05.",
        "type": "number",
        "minimum": 1e-10,
        "maximum": 0.5
      },
      "degreesOfFreedom": {
        "title": "Degrees of freedom",
        "description": "Degrees of freedom of the t or chi-square distribution, or the numerator degrees of freedom of F.",
        "type": "number",
        "minimum": 1,
        "maximum": 100000
      },
      "degreesOfFreedom2": {
        "title": "Denominator degrees of freedom",
        "description": "The second (denominator) degrees of freedom of the F distribution.",
        "type": "number",
        "minimum": 1,
        "maximum": 100000
      }
    }
  },
  "outputs": {
    "critical": {
      "label": "Critical value",
      "description": "The boundary of the rejection region: the right one for right-tailed and two-tailed tests, the left one for left-tailed tests.",
      "format": "number"
    },
    "lower": {
      "label": "Left critical value",
      "description": "For a two-tailed test, the boundary of the left part of the rejection region.",
      "format": "number"
    },
    "region": {
      "label": "Rejection region",
      "description": "The values of the test statistic that reject the null hypothesis, as intervals.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "distribution": "z",
      "testType": "two-tailed",
      "significanceLevel": 0.05,
      "degreesOfFreedom": 10,
      "degreesOfFreedom2": 10
    },
    "outputs": {
      "critical": 1.9599639845400543,
      "lower": -1.9599639845400543,
      "region": "(-∞, -1.9600) ∪ (1.9600, ∞)"
    },
    "text": "The rejection region is (-∞, -1.9600) ∪ (1.9600, ∞) (Two-tailed, Z, α = 0.05)."
  },
  "examples": [
    {
      "given": {
        "distribution": "z",
        "testType": "two-tailed",
        "significanceLevel": 0.05
      },
      "expect": {
        "critical": 1.959963984540054,
        "lower": -1.959963984540054,
        "region": "(-∞, -1.9600) ∪ (1.9600, ∞)"
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.1 (z 0.975 = 1.96); Python statistics.NormalDist().inv_cdf(0.975) = 1.959963984540054"
    },
    {
      "given": {
        "distribution": "z",
        "testType": "left-tailed",
        "significanceLevel": 0.01
      },
      "expect": {
        "critical": -2.3263478740408408,
        "region": "(-∞, -2.3263)"
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.1 (2.326); Python statistics.NormalDist().inv_cdf(0.01)"
    },
    {
      "given": {
        "distribution": "t",
        "testType": "two-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 2
      },
      "expect": {
        "critical": 4.302652729749462,
        "lower": -4.302652729749462
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.975, 2 = 4.303); closed form t = (2p − 1) ÷ √(2p(1 − p)) at p = 0.975, in content.mdx and Python"
    },
    {
      "given": {
        "distribution": "t",
        "testType": "right-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 10
      },
      "expect": {
        "critical": 1.8124611228116754
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.2 (t 0.95, 10 = 1.812); Python bisection on the closed-form t CDF, Abramowitz and Stegun 26.7.4 (docs/progress/WP-26.md)"
    },
    {
      "given": {
        "distribution": "chi-square",
        "testType": "right-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 2
      },
      "expect": {
        "critical": 5.991464547107982,
        "region": "(5.9915, ∞)"
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.4 (5.991); closed form for 2 degrees of freedom, x = −2 ln α, in content.mdx and Python"
    },
    {
      "given": {
        "distribution": "chi-square",
        "testType": "two-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 5
      },
      "expect": {
        "lower": 0.8312116134866618,
        "critical": 12.832501994030029,
        "region": "[0, 0.8312) ∪ (12.8325, ∞)"
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.4 (0.831 and 12.833); Python bisection on the closed-form chi-square CDF, Abramowitz and Stegun 26.4.4 (docs/progress/WP-26.md)"
    },
    {
      "given": {
        "distribution": "f",
        "testType": "right-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 2,
        "degreesOfFreedom2": 10
      },
      "expect": {
        "critical": 4.1028210151304005
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 2, 10 = 4.103); closed form for 2 numerator degrees of freedom, x = (d2 ÷ 2)(α^(−2/d2) − 1), in content.mdx and Python"
    },
    {
      "given": {
        "distribution": "f",
        "testType": "right-tailed",
        "significanceLevel": 0.05,
        "degreesOfFreedom": 5,
        "degreesOfFreedom2": 10
      },
      "expect": {
        "critical": 3.32583453041301
      },
      "source": "NIST/SEMATECH e-Handbook 1.3.6.7.3 (F 0.95; 5, 10 = 3.326); Python bisection on the F CDF as a finite incomplete-beta sum (docs/progress/WP-26.md)"
    }
  ],
  "sources": [],
  "related": [],
  "changelog": [
    {
      "date": "2026-09-29",
      "note": "The example notes no longer mention the old page, since cards show them (user test 2026-09-29)."
    }
  ]
}