{
  "slug": "z-score",
  "title": "Z-Score Calculator",
  "heading": "Z-Score Calculator",
  "category": "math",
  "url": "https://www.revenuelab.fyi/toolbox/z-score",
  "summary": "Standard score and percentile from a normal distribution.",
  "description": "A z-score says how many standard deviations a value sits from the mean. Converting it to a percentile tells you what share of the distribution falls below your value.",
  "formula": "z = (x − μ) ÷ σ. Percentile comes from the normal cumulative distribution.",
  "dateModified": "2026-09-30",
  "run_url": "https://www.revenuelab.fyi/api/public/calc?tool=z-score",
  "inputs": [
    {
      "id": "value",
      "label": "Your value (x)",
      "kind": "number",
      "hint": null,
      "default": 130,
      "unit": null,
      "min": null,
      "max": null
    },
    {
      "id": "mean",
      "label": "Mean (μ)",
      "kind": "number",
      "hint": null,
      "default": 100,
      "unit": null,
      "min": null,
      "max": null
    },
    {
      "id": "sd",
      "label": "Standard deviation (σ)",
      "kind": "number",
      "hint": null,
      "default": 15,
      "unit": null,
      "min": 0.0001,
      "max": null
    }
  ],
  "outputs": [
    {
      "id": "z",
      "label": "Z-score",
      "format": "decimal",
      "hint": null,
      "primary": true
    },
    {
      "id": "percentile",
      "label": "Percentile",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "above",
      "label": "Share above your value",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "twoTailed",
      "label": "Two-tailed p-value",
      "format": "percent",
      "hint": null,
      "primary": false
    },
    {
      "id": "interpretation",
      "label": "Read",
      "format": "raw",
      "hint": null,
      "primary": false
    }
  ],
  "worked_example": {
    "inputs": [
      "Your value (x): 130",
      "Mean (μ): 100",
      "Standard deviation (σ): 15"
    ],
    "outputs": [
      "Z-score: 2",
      "Percentile: 97.72%",
      "Share above your value: 2.28%",
      "Two-tailed p-value: 4.550%",
      "Read: Within the typical range"
    ]
  },
  "how_to": {
    "title": "How to use this",
    "steps": [
      "Enter your value (x).",
      "Enter mean (μ).",
      "Enter standard deviation (σ).",
      "Read your z-score on the right — it updates as you type.",
      "Hit Share to keep the scenario or send it to someone."
    ]
  },
  "scenarios": [
    {
      "name": "Conservative",
      "description": "Lower-end numbers — what if things land soft?",
      "values": {
        "value": 78,
        "mean": 60,
        "sd": 9
      }
    },
    {
      "name": "Typical",
      "description": "Defaults — the most common real-world setup.",
      "values": {
        "value": 130,
        "mean": 100,
        "sd": 15
      }
    },
    {
      "name": "Ambitious",
      "description": "Higher-end numbers — what if things really pop?",
      "values": {
        "value": 208,
        "mean": 160,
        "sd": 24
      }
    }
  ],
  "limitations": [
    "Results are rounded for display; the underlying calculation keeps full precision.",
    "Very large or very small inputs may hit floating-point limits in the browser.",
    "Inputs outside the accepted range are clamped rather than rejected."
  ],
  "faq": [
    {
      "q": "How does the z-score calculator work?",
      "a": "A z-score says how many standard deviations a value sits from the mean. Converting it to a percentile tells you what share of the distribution falls below your value. The underlying maths is: z = (x − μ) ÷ σ. Percentile comes from the normal cumulative distribution."
    },
    {
      "q": "What do I need to enter?",
      "a": "3 values: your value (x), mean (μ), and standard deviation (σ). Each field starts with a sensible default, so you can change one number at a time and watch the result move."
    },
    {
      "q": "What does the z-score result mean?",
      "a": "It is returned as a decimal number and updates live as you edit the inputs, so you can compare two or three versions of a scenario in a few seconds."
    },
    {
      "q": "Is this calculator free, and do I need an account?",
      "a": "Yes, it's free, and no account is required. Nothing you type is stored on our servers — the maths runs entirely in your browser."
    },
    {
      "q": "How accurate is the result?",
      "a": "It applies the standard formula exactly, so the arithmetic is precise. Results are rounded for display; the underlying calculation keeps full precision."
    },
    {
      "q": "Who is this tool for?",
      "a": "It's built for students, developers, and anyone who needs the answer right now without setting up a spreadsheet."
    }
  ],
  "related": [
    "https://www.revenuelab.fyi/toolbox/probability",
    "https://www.revenuelab.fyi/toolbox/standard-deviation",
    "https://www.revenuelab.fyi/toolbox/average"
  ],
  "license": "CC-BY-4.0",
  "citation": "RevenueLab — Z-Score Calculator (https://www.revenuelab.fyi/toolbox/z-score)"
}