{
    "name": "Statistics Calculator",
    "url": "https://calculators.geekmidst.com/calculators/statistics-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/statistics-calculator.md",
    "category": "Statistics & probability",
    "description": "Paste a list of numbers to get a full summary: mean, median, mode, quartiles and IQR, sample and population standard deviation, standard error and skewness.",
    "summary": "One paste gives the descriptive statistics you need to report a data set: centre, spread, quartiles and shape, with the method for each named.",
    "also_called": [
        "Descriptive statistics calculator",
        "Summary statistics 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": "mean",
            "label": "Mean",
            "format": "number"
        },
        {
            "key": "median",
            "label": "Median",
            "format": "number"
        },
        {
            "key": "mode",
            "label": "Mode",
            "format": "text"
        },
        {
            "key": "sample_sd",
            "label": "Sample standard deviation (s)",
            "format": "number"
        },
        {
            "key": "population_sd",
            "label": "Population standard deviation (σ)",
            "format": "number"
        },
        {
            "key": "sample_variance",
            "label": "Sample variance (s²)",
            "format": "number"
        },
        {
            "key": "population_variance",
            "label": "Population variance (σ²)",
            "format": "number"
        },
        {
            "key": "standard_error",
            "label": "Standard error of the mean",
            "format": "number"
        },
        {
            "key": "q1",
            "label": "First quartile (Q1)",
            "format": "number"
        },
        {
            "key": "q3",
            "label": "Third quartile (Q3)",
            "format": "number"
        },
        {
            "key": "iqr",
            "label": "Interquartile range",
            "format": "number"
        },
        {
            "key": "min",
            "label": "Minimum",
            "format": "number"
        },
        {
            "key": "max",
            "label": "Maximum",
            "format": "number"
        },
        {
            "key": "range",
            "label": "Range",
            "format": "number"
        },
        {
            "key": "count",
            "label": "Count",
            "format": "integer"
        },
        {
            "key": "sum",
            "label": "Sum",
            "format": "number"
        },
        {
            "key": "cv",
            "label": "Coefficient of variation",
            "format": "text"
        },
        {
            "key": "skewness",
            "label": "Skewness (adjusted, G₁)",
            "format": "text"
        }
    ],
    "formula": {
        "text": "Mean x̄ = Σx ÷ n;  s = √(Σ(x − x̄)² ÷ (n − 1));  σ = √(Σ(x − x̄)² ÷ n);  SE = s ÷ √n;  Quartile at position 1 + (n − 1)p in sorted data, interpolated;  G₁ = n ÷ ((n − 1)(n − 2)) × Σ((x − x̄) ÷ s)³",
        "variables": {
            "n": "number of values",
            "x̄": "mean",
            "s, σ": "sample and population standard deviation",
            "p": "0.25 for Q1, 0.5 for the median, 0.75 for Q3"
        }
    },
    "worked_example": {
        "inputs": {
            "values": "2, 4, 4, 4, 5, 5, 7, 9"
        },
        "steps": [
            "Mean = 40 ÷ 8 = 5. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32.",
            "σ = √(32 ÷ 8) = 2; s = √(32 ÷ 7) ≈ 2.1381.",
            "Q1 sits at position 1 + 7 × 0.25 = 2.75 → 4; Q3 at 6.25 → 5 + 0.25 × (7 − 5) = 5.5; IQR = 1.5."
        ],
        "result": "The data have mean 5, median 4.5, population SD 2 (sample SD 2.14) and an interquartile range of 1.5."
    },
    "interpretation": "Use the sample standard deviation when your numbers are a sample from a bigger group (the usual case) and the population figure only when you have every member. Positive skewness means a long tail of high values.",
    "assumptions": [
        "Quartiles use linear interpolation between sorted values (Hyndman–Fan type 7), the method of Excel’s QUARTILE.INC and R’s default."
    ],
    "limitations": [
        "Other quartile conventions (for example Tukey’s hinges or QUARTILE.EXC) can give slightly different Q1 and Q3 for small data sets.",
        "Skewness is the adjusted Fisher–Pearson coefficient used by Excel’s SKEW, and needs at least 3 values."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Why do different tools give different quartiles?",
            "answer": "There are at least nine published definitions. They agree for large data sets but can differ for small ones; this calculator names the one it uses so you can match it."
        },
        {
            "question": "What is the standard error for?",
            "answer": "It estimates how much the sample mean would vary from sample to sample. A 95% confidence interval for the mean is roughly mean ± 2 × SE for moderate samples."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Standard Deviation Calculator",
            "url": "https://calculators.geekmidst.com/calculators/standard-deviation-calculator"
        },
        {
            "name": "Average Calculator",
            "url": "https://calculators.geekmidst.com/calculators/average-calculator"
        },
        {
            "name": "Sample Size Calculator",
            "url": "https://calculators.geekmidst.com/calculators/sample-size-calculator"
        },
        {
            "name": "Percent Error Calculator",
            "url": "https://calculators.geekmidst.com/calculators/percent-error-calculator"
        },
        {
            "name": "Interquartile Range Calculator",
            "url": "https://calculators.geekmidst.com/calculators/interquartile-range-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}