{
  "id": "relative-standard-deviation",
  "version": "7a2d124dc255",
  "status": "published",
  "name": "RSD Calculator (Relative Standard Deviation)",
  "question": "What is the RSD of my data?",
  "summary": "Computes the relative standard deviation (RSD, %RSD) of a list of numbers: 100 times the sample or population standard deviation divided by the size of the mean.",
  "category": "statistics",
  "subcategory": "descriptive",
  "url": "https://www.acalculator.org/statistics/relative-standard-deviation-calculator",
  "markdown": "https://www.acalculator.org/statistics/relative-standard-deviation-calculator.md",
  "kind": "function",
  "method": "RSD = 100 × s ÷ |x̄| percent; s is the sample (n − 1) or population (N) standard deviation.",
  "assumptions": [
    "The RSD suits data on a ratio scale (a true zero), such as repeat measurements of a concentration.",
    "The RSD is not defined when the mean is 0.",
    "The RSD equals the coefficient of variation as a percent, except that it uses the size of the mean, so it is never negative."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "x": {
        "title": "Data values",
        "description": "The numbers, separated by commas, spaces, semicolons, or new lines.",
        "type": "array",
        "items": {
          "type": "number"
        }
      },
      "type": {
        "title": "Data are a",
        "description": "Sample: the standard deviation divides by n − 1. Population: every member is listed, so it divides by N.",
        "type": "string",
        "enum": [
          "sample",
          "population"
        ]
      }
    }
  },
  "outputs": {
    "percent": {
      "label": "Coefficient of variation (CV)",
      "description": "The standard deviation as a percent of the mean: 100 × s ÷ x̄ (negative when the mean is).",
      "format": "percent"
    },
    "rsd": {
      "label": "Relative standard deviation (RSD)",
      "description": "The standard deviation as a percent of the size of the mean: 100 × s ÷ |x̄|.",
      "format": "percent"
    },
    "cv": {
      "label": "CV as a ratio",
      "description": "s ÷ x̄, not multiplied by 100.",
      "format": "number"
    },
    "mean": {
      "label": "Mean (x̄)",
      "description": "The sum of the values divided by their count.",
      "format": "number"
    },
    "sd": {
      "label": "Standard deviation (s)",
      "description": "The sample (n − 1) or population (N) standard deviation, as chosen.",
      "format": "number"
    },
    "n": {
      "label": "Count (n)",
      "description": "How many values there are.",
      "format": "integer"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "x": "98.2, 101.5, 99.7, 100.4, 100.9, 99.3",
      "type": "sample"
    },
    "outputs": {
      "percent": 1.186591757935307,
      "rsd": 1.186591757935307,
      "cv": 0.011865917579353069,
      "mean": 99.99999999999999,
      "sd": 1.1865917579353067,
      "n": 6
    },
    "text": "The relative standard deviation of the 6 values is 1.18659%."
  },
  "examples": [
    {
      "given": {
        "x": [
          98.2,
          101.5,
          99.7,
          100.4,
          100.9,
          99.3
        ],
        "type": "sample"
      },
      "expect": {
        "rsd": 1.1865917579353067,
        "mean": 100,
        "sd": 1.1865917579353067,
        "n": 6
      },
      "source": "hand calculation in content.mdx: x̄ = 600 ÷ 6 = 100, Σ(x − x̄)² = 7.04, s = √(7.04 ÷ 5); NIST Dataplot Reference Manual, COEFFICIENT OF VARIATION (cv = s / x̄; relative standard deviation = 100 s / |x̄|), https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/coefvari.htm (retrieved 2026-10-02); NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (s² = Σ(Y − Ȳ)² ÷ (N − 1)), https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm (retrieved 2026-10-02); Python 3: statistics.stdev"
    },
    {
      "given": {
        "x": [
          10,
          12,
          23,
          23,
          16,
          23,
          21,
          16
        ],
        "type": "population"
      },
      "expect": {
        "rsd": 27.216552697590867,
        "mean": 18,
        "sd": 4.898979485566356
      },
      "source": "hand calculation in content.mdx: x̄ = 18, Σ(x − x̄)² = 192, σ = √(192 ÷ 8) = √24; NIST Dataplot Reference Manual, COEFFICIENT OF VARIATION (cv = s / x̄; relative standard deviation = 100 s / |x̄|), https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/coefvari.htm (retrieved 2026-10-02); Python 3: statistics.pstdev"
    },
    {
      "given": {
        "x": [
          -4,
          -6,
          -8
        ],
        "type": "sample"
      },
      "expect": {
        "rsd": 33.333333333333336,
        "percent": -33.333333333333336
      },
      "source": "hand calculation in content.mdx: x̄ = −6, s = 2, RSD = 100 × 2 ÷ 6; NIST Dataplot Reference Manual, COEFFICIENT OF VARIATION (cv = s / x̄; relative standard deviation = 100 s / |x̄|), https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/coefvari.htm (retrieved 2026-10-02)"
    }
  ],
  "sources": [
    "NIST Dataplot Reference Manual, COEFFICIENT OF VARIATION (cv = s ÷ x̄; the relative standard deviation is the absolute value of the coefficient of variation in percent, 100 s ÷ |x̄|). https://itl.nist.gov/div898/software/dataplot/refman2/auxillar/coefvari.htm (retrieved 2026-10-02)",
    "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.6 Measures of Scale (sample variance and standard deviation with N − 1). https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm (retrieved 2026-10-02)"
  ],
  "related": [
    "coefficient-of-variation",
    "standard-deviation",
    "variance"
  ],
  "changelog": []
}