Confidence Interval How to Calculate

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Confidence Interval Calculator

80% 90% 95% 98% 99% 99.9%

Calculation Results

Confidence Interval:

Margin of Error:

Standard Error:

Z-Score:

function calculateCI() { var mean = parseFloat(document.getElementById('sampleMean').value); var n = parseInt(document.getElementById('sampleSize').value); var sd = parseFloat(document.getElementById('stdDev').value); var level = parseFloat(document.getElementById('confLevel').value); if (isNaN(mean) || isNaN(n) || isNaN(sd) || n <= 0) { alert("Please enter valid numeric values. Sample size must be greater than zero."); return; } var zScores = { "80": 1.282, "90": 1.645, "95": 1.960, "98": 2.326, "99": 2.576, "99.9": 3.291 }; var z = zScores[level.toString()]; var standardError = sd / Math.sqrt(n); var marginOfError = z * standardError; var lowerLimit = mean – marginOfError; var upperLimit = mean + marginOfError; document.getElementById('resRange').innerHTML = lowerLimit.toFixed(4) + " to " + upperLimit.toFixed(4); document.getElementById('resMargin').innerHTML = marginOfError.toFixed(4); document.getElementById('resStdErr').innerHTML = standardError.toFixed(4); document.getElementById('resZ').innerHTML = z.toFixed(3); document.getElementById('ci-result-area').style.display = 'block'; }

How to Calculate a Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. In statistics, "confidence" refers to the probability that the procedure used to determine the interval will provide an interval that includes the true population parameter if the process were repeated many times.

The Confidence Interval Formula

For a population mean where the sample size is large (typically n > 30), the formula is:

CI = x̄ ± Z * (s / √n)
  • x̄ (Sample Mean): The average value of your data set.
  • Z (Z-Score): A constant determined by your desired confidence level (e.g., 1.96 for 95%).
  • s (Standard Deviation): The measure of dispersion in your sample.
  • n (Sample Size): The total number of observations in your sample.
  • s / √n (Standard Error): The standard deviation of the sampling distribution.

Step-by-Step Calculation Example

Imagine you are measuring the height of plants in a greenhouse. You take a sample of 100 plants (n = 100) and find their average height is 50 cm (x̄ = 50) with a standard deviation of 10 cm (s = 10). You want to calculate a 95% confidence interval.

  1. Find the Z-score: For 95% confidence, the Z-score is 1.96.
  2. Calculate Standard Error: 10 / √100 = 10 / 10 = 1.
  3. Calculate Margin of Error: 1.96 * 1 = 1.96.
  4. Determine the Range:
    • Lower: 50 – 1.96 = 48.04 cm
    • Upper: 50 + 1.96 = 51.96 cm

Result: You are 95% confident that the true average height of all plants in the greenhouse is between 48.04 cm and 51.96 cm.

Common Z-Scores

Confidence Level Z-Score
90% 1.645
95% 1.960
99% 2.576

Why Is Sample Size Important?

As the sample size (n) increases, the Standard Error decreases. This results in a narrower confidence interval, meaning your estimate is more precise. Conversely, if you want a higher level of confidence (e.g., 99% instead of 95%), the Z-score increases, which widens the interval to ensure the true mean is captured.

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