{
  "id": "percent-error",
  "version": "8daba471f55b",
  "status": "published",
  "name": "Percent Error Calculator",
  "question": "What is the percent error?",
  "summary": "Computes the percent error between an experimental (measured) value and the theoretical (true) value, or either value from the other and the percent error.",
  "category": "math",
  "subcategory": "percentages",
  "url": "https://www.acalculator.org/math/percent-error-calculator",
  "markdown": "https://www.acalculator.org/math/percent-error-calculator.md",
  "kind": "formulas",
  "method": "δ = |E − T| ÷ |T| × 100, where E is the experimental value and T the theoretical value.",
  "assumptions": [
    "The theoretical (true) value is not 0.",
    "The percent error is never negative; the signed error shows whether the measurement is high or low.",
    "Finding the experimental value from the true value and the error has two answers, one below and one above. Finding the true value from the measurement and the error also has two, E ÷ (1 + δ/100) and E ÷ (1 − δ/100), except at 100% where only the first exists."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "theoretical": {
        "title": "Theoretical value",
        "description": "The true or accepted value. It cannot be 0.",
        "type": "number"
      },
      "experimental": {
        "title": "Experimental value",
        "description": "The measured or observed value.",
        "type": "number"
      },
      "percentError": {
        "title": "Percent error",
        "description": "The size of the difference as a percent of the true value, never negative.",
        "type": "number",
        "x-unit": "percent",
        "minimum": 0
      }
    }
  },
  "solve": {
    "fillAny": 2,
    "variables": [
      "theoretical",
      "experimental",
      "percentError"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "theoretical": {
      "label": "Theoretical value",
      "description": "The true or accepted value. It cannot be 0.",
      "format": "number"
    },
    "experimental": {
      "label": "Experimental value",
      "description": "The measured or observed value.",
      "format": "number"
    },
    "error": {
      "label": "Percent error",
      "description": "The size of the difference as a percent of the true value, never negative.",
      "format": "percent"
    },
    "signed": {
      "label": "Signed error",
      "description": "The difference with its sign, (E − T) ÷ |T| × 100: positive when the measurement is too high.",
      "format": "percent"
    },
    "difference": {
      "label": "Difference",
      "description": "The experimental value minus the theoretical value, E − T.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "theoretical": 100,
      "experimental": 105
    },
    "outputs": {
      "error": 5,
      "signed": 5,
      "difference": 5
    },
    "text": "The percent error between 100 and 105 is 5%."
  },
  "examples": [
    {
      "given": {
        "theoretical": 100,
        "experimental": 105
      },
      "expect": {
        "percentError": 5,
        "signed": 5,
        "difference": 5
      },
      "source": "hand calculation in content.mdx: |105 − 100| ÷ 100 × 100 = 5%; Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
    },
    {
      "given": {
        "theoretical": 9.81,
        "experimental": 9.6
      },
      "expect": {
        "percentError": 2.1406727828746264,
        "signed": -2.1406727828746264
      },
      "source": "hand calculation in content.mdx: |9.6 − 9.81| ÷ 9.81 × 100 = 2.1407%; Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
    },
    {
      "given": {
        "theoretical": -10,
        "experimental": -12
      },
      "expect": {
        "percentError": 20,
        "signed": -20
      },
      "source": "hand calculation in content.mdx: |−12 − (−10)| ÷ |−10| × 100 = 20%; Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
    },
    {
      "given": {
        "theoretical": 100,
        "percentError": 5
      },
      "expect": {
        "experimental": 95
      },
      "source": "hand calculation in content.mdx: 100 − 5 = 95 and 100 + 5 = 105; Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
    },
    {
      "given": {
        "experimental": 105,
        "percentError": 5
      },
      "expect": {
        "theoretical": 110.52631578947368
      },
      "source": "hand calculation in content.mdx: 105 ÷ 1.05 = 100 and 105 ÷ 0.95 = 110.526; Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
    }
  ],
  "sources": [
    "Percent error: CK-12 Introductory Chemistry, section 3.13 \"Percent Error\" (Chemistry LibreTexts). https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_(CK-12)/03:_Measurements/3.13:_Percent_Error"
  ],
  "related": [
    "percentage"
  ],
  "changelog": [
    {
      "date": "2026-09-28",
      "note": "A theoretical value of 0 names the rule."
    },
    {
      "date": "2026-09-29",
      "note": "Worked examples name a published reference for their rule."
    }
  ]
}