Confidence Rate Calculator

Confidence Rate & Interval Calculator

90% Confidence 95% Confidence (Standard) 99% Confidence 99.9% Confidence

function calculateConfidence() { // Get Inputs var zScore = parseFloat(document.getElementById('crConfidenceLevel').value); var n = parseFloat(document.getElementById('crSampleSize').value); var x = parseFloat(document.getElementById('crSuccesses').value); var resultBox = document.getElementById('crResults'); // Validation if (isNaN(n) || isNaN(x) || n <= 0) { alert("Please enter a valid Sample Size greater than 0."); return; } if (x n) { alert("Number of successes (x) cannot be larger than the Sample Size (n)."); return; } // Calculation Logic (Wilson Score Interval is better for small n, but standard Wald is standard for basic calculators) // We will use the standard Wald Interval Formula: p +/- Z * sqrt(p(1-p)/n) var p = x / n; // Sample Proportion var q = 1 – p; // Standard Error Formula var se = Math.sqrt((p * q) / n); // Margin of Error var marginOfError = zScore * se; // Interval Limits var lowerLimit = p – marginOfError; var upperLimit = p + marginOfError; // Clamp results to 0-1 (0% to 100%) as probability cannot exceed these bounds if (lowerLimit 1) upperLimit = 1; // Formatting Results var pPercent = (p * 100).toFixed(2) + "%"; var moPercent = (marginOfError * 100).toFixed(2) + "%"; var seFormatted = se.toFixed(4); var lowerPercent = (lowerLimit * 100).toFixed(2) + "%"; var upperPercent = (upperLimit * 100).toFixed(2) + "%"; // Update DOM document.getElementById('resProportion').innerHTML = pPercent; document.getElementById('resZScore').innerHTML = zScore; document.getElementById('resStdError').innerHTML = seFormatted; document.getElementById('resMarginError').innerHTML = "± " + moPercent; document.getElementById('resInterval').innerHTML = "Confidence Interval: " + lowerPercent + " to " + upperPercent; // Show results resultBox.style.display = 'block'; }

Understanding Confidence Rates and Intervals

In statistics, accuracy is rarely absolute. When researchers survey a population or conduct an A/B test, they use a sample to make an educated guess about the entire population. The Confidence Rate (or Confidence Level) determines how sure you can be that your results are accurate, while the Confidence Interval provides the range in which the true value is likely to fall.

Key Metrics Definitions

• Sample Size (n): The total number of observations, people surveyed, or experiments run. Larger sample sizes generally reduce the margin of error.
• Successes (x): The count of positive outcomes (e.g., people who voted "Yes", conversions on a website, or defective units found).
• Confidence Level: The probability that if you were to repeat the experiment many times, the true population parameter would fall within the calculated interval. The industry standard is 95%.
• Margin of Error: The range of variation above and below the sample proportion. For example, if a poll shows 50% with a margin of error of 5%, the true value is likely between 45% and 55%.

The Mathematics Behind the Calculation

This calculator uses the standard Normal Approximation method (Wald Interval) to determine the confidence interval for a population proportion. The formula used is:

CI = p̂ ± Z × √[ (p̂(1 – p̂)) / n ]

Where:

• p̂ (p-hat): The sample proportion (Successes / Sample Size).
• Z: The Z-score corresponding to the chosen confidence level (e.g., 1.96 for 95%).
• n: The sample size.

Example Calculation

Imagine you run an e-commerce website and want to know your conversion rate confidence interval.

• Sample Size (n): 1,000 visitors.
• Successes (x): 50 sales.
• Confidence Level: 95% (Z = 1.96).

First, calculate the proportion (p): 50 / 1000 = 0.05 (5%).

Next, calculate the Standard Error: √[(0.05 × 0.95) / 1000] ≈ 0.0069.

Finally, calculate the Margin of Error: 1.96 × 0.0069 ≈ 0.0135 (1.35%).

Result: You can be 95% confident that your true conversion rate is between 3.65% and 6.35%.

Why Use 95% Confidence?

95% is the most common standard in scientific research and polling. It offers a balanced trade-off between precision and certainty. Choosing a 99% confidence level increases your certainty but widens the interval, making the result less precise. Conversely, a 90% confidence level creates a narrower (more precise) interval but increases the risk that the true value falls outside of it.