{
    "name": "Normal Distribution CDF Calculator",
    "url": "https://calculators.geekmidst.com/calculators/normal-distribution-cdf-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/normal-distribution-cdf-calculator.md",
    "category": "Statistics & probability",
    "description": "Find the area under any normal curve: P(X ≤ x), P(X ≥ x), between two values or outside them, for your mean and standard deviation.",
    "summary": "Enter a normal distribution’s mean and standard deviation and the values you care about to get the probability below, above, between or outside them.",
    "also_called": [
        "Normal CDF calculator",
        "Bell curve probability calculator"
    ],
    "applies_to": "Worldwide",
    "last_updated": "2026-10-29",
    "computation": "Results are computed in the visitor’s browser on the HTML page. This API describes the calculator; it does not compute results.",
    "inputs": [
        {
            "name": "mean",
            "label": "Mean (μ)",
            "type": "number",
            "default": "100"
        },
        {
            "name": "sd",
            "label": "Standard deviation (σ)",
            "type": "number",
            "default": "15"
        },
        {
            "name": "mode",
            "label": "Probability of",
            "type": "select",
            "default": "between",
            "options": {
                "below": "Below x: P(X ≤ x)",
                "above": "Above x: P(X ≥ x)",
                "between": "Between a and b",
                "outside": "Outside a and b (both tails)"
            }
        },
        {
            "name": "x",
            "label": "Value (x)",
            "type": "number",
            "default": "130",
            "shown_when": {
                "mode": [
                    "below",
                    "above"
                ]
            }
        },
        {
            "name": "a",
            "label": "Lower value (a)",
            "type": "number",
            "default": "85",
            "shown_when": {
                "mode": [
                    "between",
                    "outside"
                ]
            }
        },
        {
            "name": "b",
            "label": "Upper value (b)",
            "type": "number",
            "default": "115",
            "shown_when": {
                "mode": [
                    "between",
                    "outside"
                ]
            }
        }
    ],
    "outputs": [
        {
            "key": "probability",
            "label": "Probability",
            "format": "number"
        },
        {
            "key": "percent",
            "label": "As a percentage",
            "format": "percent"
        },
        {
            "key": "p_text",
            "label": "Probability, full precision",
            "format": "text"
        },
        {
            "key": "z",
            "label": "Standardised value(s)",
            "format": "text"
        },
        {
            "key": "event",
            "label": "Event",
            "format": "text"
        }
    ],
    "formula": {
        "text": "z = (x − μ) ÷ σ;  P(X ≤ x) = Φ(z) = ½·erfc(−z ÷ √2);  P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)",
        "variables": {
            "μ, σ": "mean and standard deviation",
            "Φ": "standard normal cumulative distribution function",
            "erfc": "complementary error function, computed to near machine precision (also far into the tails)"
        }
    },
    "worked_example": {
        "inputs": {
            "mean": "100",
            "sd": "15",
            "mode": "between",
            "x": "130",
            "a": "85",
            "b": "115"
        },
        "steps": [
            "z for 85 = (85 − 100) ÷ 15 = −1; z for 115 = +1.",
            "Φ(1) = 0.841345 and Φ(−1) = 0.158655.",
            "P(85 ≤ X ≤ 115) = 0.841345 − 0.158655 = 0.682689."
        ],
        "result": "For scores with mean 100 and standard deviation 15, 68.27% fall between 85 and 115."
    },
    "interpretation": "For a continuous distribution, “below” and “at or below” are the same: the probability of hitting a value exactly is zero. The answer is only as good as the normal model, so check a histogram first if the data could be skewed or have heavy tails.",
    "assumptions": [
        "The quantity follows a normal distribution with the mean and standard deviation entered."
    ],
    "limitations": [
        "Bounded or skewed data (times, incomes, counts near zero) are often poorly described by a normal curve, especially in the tails."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "How is this different from the z-score calculator?",
            "answer": "The z-score calculator works from one value and symmetric bands around the mean. This one handles any interval between two values and either tail."
        },
        {
            "question": "Why does the empirical rule say 68%?",
            "answer": "Because the area between z = −1 and z = 1 is 0.6827. Between ±2 it is 0.9545, and between ±3 it is 0.9973."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Normal Distribution Quantile Calculator",
            "url": "https://calculators.geekmidst.com/calculators/normal-distribution-quantile-calculator"
        },
        {
            "name": "Z-score Calculator",
            "url": "https://calculators.geekmidst.com/calculators/z-score-calculator"
        },
        {
            "name": "P-value Calculator",
            "url": "https://calculators.geekmidst.com/calculators/p-value-calculator"
        },
        {
            "name": "Standard Deviation Calculator",
            "url": "https://calculators.geekmidst.com/calculators/standard-deviation-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}