{
    "name": "Median Absolute Deviation Calculator",
    "url": "https://calculators.geekmidst.com/calculators/median-absolute-deviation-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/median-absolute-deviation-calculator.md",
    "category": "Statistics & probability",
    "description": "Compute the median absolute deviation, its normal-consistent scaled version (×1.4826) and modified z-score outliers from a list of numbers.",
    "summary": "MAD is the median distance of the data from their median. It measures spread like a standard deviation but is barely moved by outliers.",
    "also_called": [
        "MAD 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": "values",
            "label": "Numbers",
            "type": "textarea",
            "default": "2, 4, 4, 4, 5, 5, 7, 9"
        }
    ],
    "outputs": [
        {
            "key": "mad",
            "label": "Median absolute deviation (raw)",
            "format": "number"
        },
        {
            "key": "scaled_mad",
            "label": "Scaled MAD (estimates σ for normal data)",
            "format": "number"
        },
        {
            "key": "median",
            "label": "Median",
            "format": "number"
        },
        {
            "key": "mean_absolute_deviation",
            "label": "Mean absolute deviation from the mean",
            "format": "number"
        },
        {
            "key": "outliers",
            "label": "Outliers (|modified z| > 3.5)",
            "format": "text"
        },
        {
            "key": "count",
            "label": "Count",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "MAD = median(|xᵢ − median(x)|);  scaled MAD = 1.4826 × MAD;  modified z = 0.6745 × (xᵢ − median) ÷ MAD",
        "variables": {
            "xᵢ": "each value",
            "1.4826": "1 ÷ Φ⁻¹(0.75), which makes MAD estimate σ for normal data",
            "3.5": "Iglewicz–Hoaglin outlier cut-off for modified z"
        }
    },
    "worked_example": {
        "inputs": {
            "values": "2, 4, 4, 4, 5, 5, 7, 9"
        },
        "steps": [
            "Median of the sorted data = (4 + 5) ÷ 2 = 4.5.",
            "Absolute deviations: 2.5, 0.5, 0.5, 0.5, 0.5, 0.5, 2.5, 4.5; their median is 0.5.",
            "Scaled MAD = 1.4826 × 0.5 ≈ 0.7413 (R’s mad() default)."
        ],
        "result": "The MAD is 0.5, or 0.7413 scaled to the standard-deviation scale."
    },
    "interpretation": "Compare the scaled MAD with the standard deviation: if the SD is much larger, a few extreme values are inflating it. Values with a modified z-score above 3.5 deserve a closer look.",
    "assumptions": [
        "The scale factor 1.4826 assumes the bulk of the data is roughly normal.",
        "Outliers are flagged with the Iglewicz–Hoaglin modified z-score rule."
    ],
    "limitations": [
        "If more than half the values are identical, MAD is 0 and modified z-scores cannot be computed."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Is MAD the same as mean absolute deviation?",
            "answer": "No. Median absolute deviation uses medians throughout and resists outliers; mean absolute deviation averages distances from the mean. Both are shown."
        },
        {
            "question": "Which software matches this?",
            "answer": "R’s mad(x) returns the scaled value; mad(x, constant = 1) returns the raw MAD. Excel has no built-in MAD."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Standard Deviation Calculator",
            "url": "https://calculators.geekmidst.com/calculators/standard-deviation-calculator"
        },
        {
            "name": "Interquartile Range Calculator",
            "url": "https://calculators.geekmidst.com/calculators/interquartile-range-calculator"
        },
        {
            "name": "Tukey Fence Outlier Calculator",
            "url": "https://calculators.geekmidst.com/calculators/outlier-tukey-fence-calculator"
        },
        {
            "name": "Statistics Calculator",
            "url": "https://calculators.geekmidst.com/calculators/statistics-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}