{
    "name": "RMSE Calculator",
    "url": "https://calculators.geekmidst.com/calculators/root-mean-square-error-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/root-mean-square-error-calculator.md",
    "category": "Statistics & probability",
    "description": "Paste actual and predicted values to get the root mean square error, MSE, MAE, mean bias and RMSE normalised by the mean and by the range.",
    "summary": "RMSE is the typical size of a prediction error in the units of the data, with large errors weighted more heavily because they are squared before averaging.",
    "also_called": [
        "Root mean squared error calculator",
        "MSE 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": "actual",
            "label": "Actual (observed) values",
            "type": "textarea",
            "default": "3, 5, 7, 9"
        },
        {
            "name": "predicted",
            "label": "Predicted (forecast) values, same order",
            "type": "textarea",
            "default": "2.5, 5.5, 7, 8"
        }
    ],
    "outputs": [
        {
            "key": "rmse",
            "label": "RMSE",
            "format": "number"
        },
        {
            "key": "mse",
            "label": "MSE (mean squared error)",
            "format": "number"
        },
        {
            "key": "mae",
            "label": "MAE (mean absolute error)",
            "format": "number"
        },
        {
            "key": "bias",
            "label": "Mean error (actual − predicted)",
            "format": "number"
        },
        {
            "key": "nrmse_mean",
            "label": "RMSE ÷ mean of actual values",
            "format": "text"
        },
        {
            "key": "nrmse_range",
            "label": "RMSE ÷ range of actual values",
            "format": "text"
        },
        {
            "key": "n",
            "label": "Pairs (n)",
            "format": "integer"
        }
    ],
    "formula": {
        "text": "eᵢ = actualᵢ − predictedᵢ;  MSE = Σeᵢ² ÷ n;  RMSE = √MSE",
        "variables": {
            "eᵢ": "error for pair i",
            "n": "number of pairs (the divisor is n, not n − 1 or n − k)"
        }
    },
    "worked_example": {
        "inputs": {
            "actual": "3, 5, 7, 9",
            "predicted": "2.5, 5.5, 7, 8"
        },
        "steps": [
            "Errors: 0.5, −0.5, 0, 1. Squares: 0.25, 0.25, 0, 1.",
            "MSE = 1.5 ÷ 4 = 0.375.",
            "RMSE = √0.375 = 0.6124."
        ],
        "result": "The predictions are off by about 0.61 units in the root-mean-square sense, slightly more than the 0.5 mean absolute error because of the one larger miss."
    },
    "interpretation": "RMSE is always at least as large as MAE; a big gap between them means a few large errors. Compare RMSE only between models predicting the same quantity in the same units, or use the normalised versions. A non-zero mean error shows systematic over- or under-prediction.",
    "assumptions": [
        "Each predicted value is paired with the actual value in the same position.",
        "RMSE here divides by n (descriptive). The residual standard error of a regression divides by n − k − 1 instead."
    ],
    "limitations": [
        "Sensitive to outliers by design; report MAE alongside it if a few large errors should not dominate."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "RMSE or MAE?",
            "answer": "RMSE penalises large errors more, which suits costs that grow faster than the error. MAE treats every unit of error equally and is easier to explain."
        },
        {
            "question": "Is a lower RMSE always better?",
            "answer": "For the same data, yes. Across different data sets or units the raw numbers are not comparable."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Mean Absolute Error Calculator",
            "url": "https://calculators.geekmidst.com/calculators/mean-absolute-error-calculator"
        },
        {
            "name": "MAPE Calculator",
            "url": "https://calculators.geekmidst.com/calculators/mean-absolute-percentage-error-calculator"
        },
        {
            "name": "R-Squared Calculator",
            "url": "https://calculators.geekmidst.com/calculators/regression-r-squared-calculator"
        },
        {
            "name": "Standard Deviation Calculator",
            "url": "https://calculators.geekmidst.com/calculators/standard-deviation-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}