{
  "id": "empirical-rule",
  "version": "17e7d6582853",
  "status": "published",
  "name": "Empirical Rule Calculator",
  "question": "What does the empirical rule say?",
  "summary": "Computes the ranges within 1, 2, and 3 standard deviations of the mean of a list of numbers, and the percent of the numbers in each, to compare with the 68-95-99.7 rule.",
  "category": "statistics",
  "subcategory": "descriptive",
  "url": "https://www.acalculator.org/statistics/empirical-rule-calculator",
  "markdown": "https://www.acalculator.org/statistics/empirical-rule-calculator.md",
  "kind": "function",
  "method": "Find the mean and standard deviation; for k = 1, 2, 3 the range is mean ± k × SD, and the share is the count of numbers with |x − mean| ≤ k × SD, divided by n.",
  "assumptions": [
    "The population standard deviation (divide by n) is the default, as on the old page. Pick sample (divide by n − 1) for a sample.",
    "A number on the end of a range counts as inside it, within a relative 2^-49 of |mean| + k × SD for rounding.",
    "The 68%, 95%, and 99.7% figures hold for a normal distribution. Other shapes can differ a lot."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "input": {
        "title": "Your numbers",
        "description": "The numbers, separated by commas, spaces, semicolons, or new lines. At least 2.",
        "type": "array",
        "items": {
          "type": "number"
        }
      },
      "sd": {
        "title": "Standard deviation",
        "description": "Population (divide by n) for a whole group, or sample (divide by n − 1) for a sample of a larger group.",
        "type": "string",
        "enum": [
          "population",
          "sample"
        ]
      }
    }
  },
  "outputs": {
    "share1": {
      "label": "Within 1 standard deviation",
      "description": "The percent of your numbers within 1 standard deviation of the mean. A normal distribution has about 68%.",
      "format": "percent"
    },
    "share2": {
      "label": "Within 2 standard deviations",
      "description": "The percent of your numbers within 2 standard deviations of the mean. A normal distribution has about 95%.",
      "format": "percent"
    },
    "share3": {
      "label": "Within 3 standard deviations",
      "description": "The percent of your numbers within 3 standard deviations of the mean. A normal distribution has about 99.7%.",
      "format": "percent"
    },
    "mean": {
      "label": "Mean (μ)",
      "description": "The sum divided by the count.",
      "format": "number"
    },
    "sd": {
      "label": "Standard deviation",
      "description": "The population or sample standard deviation, as chosen.",
      "format": "number"
    },
    "low1": {
      "label": "Mean − 1 SD",
      "description": "The low end of the range within 1 standard deviation of the mean.",
      "format": "number"
    },
    "high1": {
      "label": "Mean + 1 SD",
      "description": "The high end of the range within 1 standard deviation of the mean.",
      "format": "number"
    },
    "count1": {
      "label": "Numbers within 1 SD",
      "description": "How many of your numbers lie within 1 standard deviation of the mean, ends included.",
      "format": "integer"
    },
    "low2": {
      "label": "Mean − 2 SD",
      "description": "The low end of the range within 2 standard deviations of the mean.",
      "format": "number"
    },
    "high2": {
      "label": "Mean + 2 SD",
      "description": "The high end of the range within 2 standard deviations of the mean.",
      "format": "number"
    },
    "count2": {
      "label": "Numbers within 2 SD",
      "description": "How many of your numbers lie within 2 standard deviations of the mean, ends included.",
      "format": "integer"
    },
    "low3": {
      "label": "Mean − 3 SD",
      "description": "The low end of the range within 3 standard deviations of the mean.",
      "format": "number"
    },
    "high3": {
      "label": "Mean + 3 SD",
      "description": "The high end of the range within 3 standard deviations of the mean.",
      "format": "number"
    },
    "count3": {
      "label": "Numbers within 3 SD",
      "description": "How many of your numbers lie within 3 standard deviations of the mean, ends included.",
      "format": "integer"
    },
    "count": {
      "label": "Count",
      "description": "How many numbers are in the list.",
      "format": "integer"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "input": "1, 2, 3, 4, 5, 6, 7, 8, 9, 10",
      "sd": "population"
    },
    "outputs": {
      "share1": 60,
      "share2": 100,
      "share3": 100,
      "mean": 5.5,
      "sd": 2.8722813232690143,
      "low1": 2.6277186767309857,
      "high1": 8.372281323269014,
      "count1": 6,
      "low2": -0.24456264653802862,
      "high2": 11.244562646538029,
      "count2": 10,
      "low3": -3.116843969807043,
      "high3": 14.116843969807043,
      "count3": 10,
      "count": 10
    },
    "text": "60% of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 lie within 1 standard deviation of the mean (2.627719 to 8.372281), 100% within 2, and 100% within 3."
  },
  "examples": [
    {
      "given": {
        "input": [
          2,
          4,
          4,
          4,
          5,
          5,
          7,
          9
        ],
        "sd": "population"
      },
      "expect": {
        "mean": 5,
        "sd": 2,
        "low1": 3,
        "high1": 7,
        "count1": 6,
        "share1": 75,
        "low2": 1,
        "high2": 9,
        "count2": 8,
        "share2": 100,
        "low3": -1,
        "high3": 11,
        "share3": 100
      },
      "source": "hand calculation in content.mdx: 4, 4, 4, 5, 5, 7 lie in [3, 7] (6 of 8 = 75%); all 8 lie in [1, 9]; OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution"
    },
    {
      "given": {
        "input": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10
        ],
        "sd": "population"
      },
      "expect": {
        "mean": 5.5,
        "sd": 2.8722813232690143,
        "low1": 2.6277186767309857,
        "high1": 8.372281323269014,
        "count1": 6,
        "share1": 60,
        "share2": 100,
        "share3": 100
      },
      "source": "hand calculation in content.mdx: σ = √8.25; 3 to 8 lie in [2.63, 8.37], 6 of 10 = 60%. Python statistics.pstdev(range(1, 11)) = 2.8722813232690143; OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution"
    },
    {
      "given": {
        "input": [
          1,
          2,
          3,
          4,
          5,
          6,
          7,
          8,
          9,
          10
        ],
        "sd": "sample"
      },
      "expect": {
        "sd": 3.0276503540974917,
        "low1": 2.4723496459025083,
        "high1": 8.527650354097492,
        "count1": 6,
        "share1": 60
      },
      "source": "hand calculation in content.mdx: s = √(82.5 ÷ 9); Python statistics.stdev(range(1, 11)) = 3.0276503540974917; OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution"
    }
  ],
  "sources": [
    "OpenStax, Introductory Statistics 2e, §6.1 The Standard Normal Distribution (the empirical rule: about 68%, 95% and 99.7%). https://openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution",
    "OpenStax, Introductory Statistics 2e, §2.7 Measures of the Spread of the Data (standard deviation and variance). https://openstax.org/books/introductory-statistics-2e/pages/2-7-measures-of-the-spread-of-the-data"
  ],
  "related": [],
  "changelog": [
    {
      "date": "2026-09-29",
      "note": "Worked examples name a published reference for their rule."
    }
  ]
}