{
    "name": "Statistical Power Calculator",
    "url": "https://calculators.geekmidst.com/calculators/statistical-power-approximation-calculator",
    "markdown": "https://calculators.geekmidst.com/calculators/statistical-power-approximation-calculator.md",
    "category": "Statistics & probability",
    "description": "Approximate the power of a test of means from effect size d, sample size and α, and the sample size needed for a target power, one- or two-sided.",
    "summary": "Power is the chance a study detects an effect that really exists. Enter Cohen’s d, the sample size and α to see the power, or the sample needed to reach, say, 80%.",
    "also_called": [
        "Power analysis 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": "design",
            "label": "Design",
            "type": "segmented",
            "default": "two",
            "options": {
                "two": "Two independent groups",
                "one": "One sample or paired"
            }
        },
        {
            "name": "d",
            "label": "Effect size (Cohen’s d)",
            "type": "number",
            "default": "0.5"
        },
        {
            "name": "n",
            "label": "Sample size (per group, or pairs)",
            "type": "number",
            "default": "64"
        },
        {
            "name": "alpha",
            "label": "Significance level α",
            "type": "number",
            "default": "5",
            "unit": "%"
        },
        {
            "name": "sides",
            "label": "Test",
            "type": "segmented",
            "default": "two",
            "options": {
                "two": "Two-sided",
                "one": "One-sided"
            }
        },
        {
            "name": "target",
            "label": "Target power",
            "type": "number",
            "default": "80",
            "unit": "%"
        }
    ],
    "outputs": [
        {
            "key": "power",
            "label": "Power at this sample size",
            "format": "percent"
        },
        {
            "key": "n_t_test",
            "label": "n for target power (t-test, Guenther-adjusted)",
            "format": "integer"
        },
        {
            "key": "n_required",
            "label": "n for target power (z approximation)",
            "format": "integer"
        },
        {
            "key": "total_required",
            "label": "Total participants (t-test)",
            "format": "integer"
        },
        {
            "key": "noncentrality",
            "label": "Noncentrality d√(n/k)",
            "format": "number"
        },
        {
            "key": "critical_z",
            "label": "Critical z",
            "format": "number"
        }
    ],
    "formula": {
        "text": "δ = d√(n/k), k = 2 for two groups, 1 for one sample;  two-sided power = Φ(δ − z₁₋α/₂) + Φ(−δ − z₁₋α/₂);  n = k(z_α + z_β)² ÷ d²;  t-adjusted n = n + z_α² ÷ (2k)",
        "variables": {
            "d": "standardised effect size",
            "n": "per-group size (or number of pairs)",
            "z_α": "critical value (z₁₋α/₂ two-sided, z₁₋α one-sided)",
            "z_β": "normal quantile of the target power"
        }
    },
    "worked_example": {
        "inputs": {
            "design": "two",
            "d": "0.5",
            "n": "64",
            "alpha": "5",
            "sides": "two",
            "target": "80"
        },
        "steps": [
            "δ = 0.5 × √(64 ÷ 2) = 2.8284; power ≈ Φ(2.8284 − 1.96) = Φ(0.8685) = 80.7%.",
            "n = 2 × (1.96 + 0.8416)² ÷ 0.25 = 62.8 → 63 per group (z).",
            "Guenther adjustment: 62.8 + 1.96² ÷ 4 = 63.75 → 64 per group, matching Cohen’s table for a t-test."
        ],
        "result": "Detecting d = 0.5 with 80% power at α = 0.05 (two-sided) needs about 64 people per group, 128 in total."
    },
    "interpretation": "Plan for the smallest effect that would matter in practice, not the effect you hope for. Halving d roughly quadruples the sample needed.",
    "assumptions": [
        "Normal approximation to the t-test; the exact noncentral-t power is slightly lower for small samples (0.8015 rather than 0.8074 in the example).",
        "Equal group sizes and a common standard deviation."
    ],
    "limitations": [
        "Not for proportions (use the A/B test sample size calculator), ANOVA, regression or clustered designs.",
        "Power analysis is only as good as the assumed effect size."
    ],
    "rules": null,
    "faqs": [
        {
            "question": "Why 80% power?",
            "answer": "It is a common convention balancing cost against a 20% risk of missing a real effect. Many trials now target 90%."
        },
        {
            "question": "Should I use a one-sided test to save participants?",
            "answer": "Only if an effect in the other direction would be treated exactly like no effect, and you decided before collecting data."
        }
    ],
    "sources": [],
    "related": [
        {
            "name": "Cohen’s d Effect Size Calculator",
            "url": "https://calculators.geekmidst.com/calculators/cohen-d-effect-size-calculator"
        },
        {
            "name": "A/B Test Sample Size Calculator",
            "url": "https://calculators.geekmidst.com/calculators/ab-test-sample-size-calculator"
        },
        {
            "name": "Sample Size Calculator",
            "url": "https://calculators.geekmidst.com/calculators/sample-size-calculator"
        },
        {
            "name": "P-value Calculator",
            "url": "https://calculators.geekmidst.com/calculators/p-value-calculator"
        }
    ],
    "disclaimer": null,
    "publisher": "GM¹X CalcVerse"
}