{
  "id": "point-estimate",
  "version": "8808bb626ff2",
  "status": "published",
  "name": "Point Estimate Calculator",
  "question": "What is the point estimate?",
  "summary": "Computes the point estimate of a proportion (successes ÷ trials) with its Wilson confidence interval, or of a mean from data or a summary with its t confidence interval and margin of error.",
  "category": "statistics",
  "subcategory": "tests",
  "url": "https://www.acalculator.org/statistics/point-estimate-calculator",
  "markdown": "https://www.acalculator.org/statistics/point-estimate-calculator.md",
  "kind": "function",
  "method": "Proportion: p̂ = x ÷ n; Wilson limits (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n). Mean: x̄ ± t(1 − α/2, N − 1) × s ÷ √N.",
  "assumptions": [
    "The sample is random and the trials or values are independent.",
    "The t interval assumes the values come from a roughly normal population, or a large sample.",
    "z and t are the two-sided critical values for the confidence level: α = 1 − level, upper quantile α/2."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "mode": {
        "title": "Estimate",
        "description": "A proportion from successes and trials, or a mean from data or from a summary.",
        "type": "string",
        "enum": [
          "proportion",
          "data",
          "summary"
        ]
      },
      "x": {
        "title": "Successes (x)",
        "description": "How many trials had the outcome of interest, such as defects found.",
        "type": "integer",
        "minimum": 0,
        "maximum": 1000000000
      },
      "n": {
        "title": "Trials (n)",
        "description": "The sample size: how many trials or items were checked.",
        "type": "integer",
        "minimum": 1,
        "maximum": 1000000000
      },
      "data": {
        "title": "Data",
        "description": "The sample values, separated by spaces, commas or new lines; at least 2.",
        "type": "array",
        "items": {
          "type": "number"
        }
      },
      "N": {
        "title": "Sample size (N)",
        "description": "How many values are in the sample; at least 2.",
        "type": "integer",
        "minimum": 2,
        "maximum": 1000000000
      },
      "mean": {
        "title": "Sample mean (x̄)",
        "description": "The average of the sample.",
        "type": "number",
        "minimum": -1000000000000,
        "maximum": 1000000000000
      },
      "sd": {
        "title": "Sample standard deviation (s)",
        "description": "The standard deviation of the sample (n − 1 in the denominator).",
        "type": "number",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "cl": {
        "title": "Confidence level",
        "description": "The confidence level of the interval, such as 95.",
        "type": "number",
        "x-unit": "percent",
        "minimum": 50,
        "maximum": 99.99
      }
    }
  },
  "outputs": {
    "estimate": {
      "label": "Point estimate",
      "description": "The sample proportion p̂ = x ÷ n, or the sample mean x̄.",
      "format": "number"
    },
    "lower": {
      "label": "Lower limit",
      "description": "The lower end of the confidence interval (Wilson for a proportion, t for a mean).",
      "format": "number"
    },
    "upper": {
      "label": "Upper limit",
      "description": "The upper end of the confidence interval.",
      "format": "number"
    },
    "margin": {
      "label": "Margin of error",
      "description": "Half the width of the interval: (upper − lower) ÷ 2.",
      "format": "number"
    },
    "center": {
      "label": "Wilson center",
      "description": "The middle of the Wilson interval: (p̂ + z² ÷ 2n) ÷ (1 + z² ÷ n), pulled toward 1/2.",
      "format": "number"
    },
    "se": {
      "label": "Standard error",
      "description": "√(p̂(1 − p̂) ÷ n) for a proportion, s ÷ √N for a mean.",
      "format": "number"
    },
    "sd": {
      "label": "Standard deviation (s)",
      "description": "The sample standard deviation.",
      "format": "number"
    },
    "critical": {
      "label": "Critical value",
      "description": "z (proportion) or t with N − 1 degrees of freedom (mean) for the confidence level.",
      "format": "number"
    },
    "count": {
      "label": "Sample size",
      "description": "The number of trials or values in the sample.",
      "format": "integer"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "mode": "proportion",
      "x": 26,
      "n": 200,
      "data": "12.1 11.8 12.4 12.0 12.2",
      "N": 195,
      "mean": 9.26146,
      "sd": 0.022789,
      "cl": 95
    },
    "outputs": {
      "estimate": 0.13,
      "lower": 0.0902820227966932,
      "upper": 0.18366351872248413,
      "margin": 0.046690747962895456,
      "center": 0.13697277075958866,
      "se": 0.02378024390118823,
      "critical": 1.9599639845400554,
      "count": 200
    },
    "text": "The point estimate is 0.13, with a 95% confidence interval from 0.0902820228 to 0.1836635187."
  },
  "examples": [
    {
      "given": {
        "mode": "proportion",
        "x": 26,
        "n": 200,
        "cl": 90
      },
      "expect": {
        "estimate": 0.13,
        "lower": 0.09577274821084578
      },
      "source": "NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (p̂ = N_d ÷ N; the Wilson score limits). https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm (p̂ = 0.13, N = 200: p ≥ 0.09577)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "mode": "summary",
        "N": 195,
        "mean": 9.26146,
        "sd": 0.022789,
        "cl": 95
      },
      "expect": {
        "estimate": 9.26146,
        "lower": 9.258241349743406,
        "upper": 9.264678650256593
      },
      "source": "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ × s ÷ √N). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (N = 195, mean 9.261460, s = 0.022789, t = 1.9723: interval 9.258242 to 9.264679)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "mode": "data",
        "data": [
          12.1,
          11.8,
          12.4,
          12,
          12.2
        ],
        "cl": 95
      },
      "expect": {
        "estimate": 12.1,
        "sd": 0.22360679774997896,
        "margin": 0.27764451051977956
      },
      "source": "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ × s ÷ √N). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm; hand calculation in content.mdx: x̄ = 60.5 ÷ 5 = 12.1; s = √(0.2 ÷ 4) = 0.2236; t₀.₉₇₅,₄ = 2.776445",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (point estimate p̂ = N_d ÷ N; normal approximation interval; Wilson score limits; example p̂ = 0.13, N = 200, p ≥ 0.09577). https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm (retrieved 2026-10-02)",
    "NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ s ÷ √N; example N = 195, mean 9.261460, s = 0.022789, interval 9.258242 to 9.264679). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (retrieved 2026-10-02)"
  ],
  "related": [
    "confidence-interval",
    "standard-error",
    "sample-size",
    "mean-median-mode"
  ],
  "changelog": []
}