{
  "id": "inverse-normal",
  "version": "aee0535df0b2",
  "status": "published",
  "name": "Inverse Normal Calculator (invNorm)",
  "question": "InvNorm calculator: x from an area",
  "summary": "Finds the value x with a given area under a normal curve to its left, to its right, or in the center (invNorm), with the z-score and a chart of the shaded area.",
  "category": "statistics",
  "subcategory": "distributions",
  "url": "https://www.acalculator.org/statistics/inverse-normal-calculator",
  "markdown": "https://www.acalculator.org/statistics/inverse-normal-calculator.md",
  "kind": "function",
  "method": "x = μ + σz. Left: Φ(z) = area. Right: 1 − Φ(z) = area. Center: Φ(z) − Φ(−z) = area, with ends μ ± σz. Φ is the standard normal CDF.",
  "assumptions": [
    "The values follow a normal distribution with the mean and standard deviation you type.",
    "The area is a probability, more than 0 and less than 1 (at least 10⁻³⁰⁰).",
    "Left and right match invNorm(area, μ, σ, LEFT) and RIGHT; center splits the rest of the area evenly between the two tails."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "p": {
        "title": "Area (probability)",
        "description": "The area under the normal curve, more than 0 and less than 1, such as 0.95.",
        "type": "number",
        "minimum": 1e-300,
        "exclusiveMaximum": 1
      },
      "tail": {
        "title": "Area is",
        "description": "Where the area lies: to the left of x, to the right of x, or in the center around the mean.",
        "type": "string",
        "enum": [
          "left",
          "right",
          "center"
        ]
      },
      "mean": {
        "title": "Mean (μ)",
        "description": "The mean of the normal distribution; 0 for the standard normal.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "sd": {
        "title": "Standard deviation (σ)",
        "description": "The standard deviation, more than 0; 1 for the standard normal.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      }
    }
  },
  "outputs": {
    "x": {
      "label": "x",
      "description": "The value with the area to its left or right; for a center area, the upper end μ + zσ.",
      "format": "number"
    },
    "z": {
      "label": "z-score",
      "description": "How many standard deviations x is from the mean: (x − μ) ÷ σ.",
      "format": "number"
    },
    "lower": {
      "label": "Lower end",
      "description": "For a center area: μ − zσ, the low end of the middle area.",
      "format": "number"
    },
    "upper": {
      "label": "Upper end",
      "description": "For a center area: μ + zσ, the high end of the middle area.",
      "format": "number"
    },
    "region": {
      "label": "The area lies",
      "description": "Below x, above x, or between the two ends.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "p": 0.95,
      "tail": "left",
      "mean": 0,
      "sd": 1
    },
    "outputs": {
      "x": 1.6448536269514724,
      "z": 1.6448536269514724,
      "region": "below 1.644853627"
    },
    "text": "An area of 0.95 under this normal curve lies below 1.644853627."
  },
  "examples": [
    {
      "given": {
        "p": 0.95,
        "tail": "left",
        "mean": 0,
        "sd": 1
      },
      "expect": {
        "x": 1.6448536269514715,
        "z": 1.6448536269514715
      },
      "source": "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area from 0 to z: 0.44950 at z = 1.64, 0.45053 at 1.65, 0.47500 at 1.96). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (1.6449 lies between 1.64 and 1.65); NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm; Python 3: statistics.NormalDist().inv_cdf(0.95) (tolerance 1e-12: Python’s inv_cdf is accurate to about 1e-15)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "p": 0.95,
        "tail": "center",
        "mean": 0,
        "sd": 1
      },
      "expect": {
        "lower": -1.9599639845400534,
        "upper": 1.9599639845400534,
        "region": "between −1.959963985 and 1.959963985"
      },
      "source": "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area from 0 to z: 0.44950 at z = 1.64, 0.45053 at 1.65, 0.47500 at 1.96). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (the area from 0 to 1.96 is 0.475, so the middle 95% runs from −1.96 to 1.96); Python 3: statistics.NormalDist().inv_cdf(0.975)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "p": 0.1,
        "tail": "right",
        "mean": 100,
        "sd": 15
      },
      "expect": {
        "x": 119.22327348316901,
        "z": 1.2815515655446006
      },
      "source": "hand calculation in content.mdx: x = 100 + 15 × 1.28155 = 119.2233; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm; Python 3: statistics.NormalDist(100, 15).inv_cdf(0.9)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "p": 0.5,
        "tail": "left",
        "mean": 7,
        "sd": 2
      },
      "expect": {
        "x": 7,
        "z": 0
      },
      "source": "hand calculation in content.mdx: half the area lies below the mean; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm"
    }
  ],
  "sources": [
    "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution (the percent point function, the inverse of the CDF, has no closed form and is computed numerically). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm (retrieved 2026-10-02)",
    "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7.1 Cumulative Distribution Function of the Standard Normal Distribution (table of the area under the standard normal curve from 0 to z; add 0.5 for the area below z). https://www.itl.nist.gov/div898/handbook/eda/section3/eda3671.htm (retrieved 2026-10-02)"
  ],
  "related": [
    "normal-distribution",
    "z-score",
    "critical-value",
    "p-value"
  ],
  "changelog": []
}