A confidence interval (CI) is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. It quantifies the uncertainty associated with estimating a population parameter from a sample. A higher confidence level means a wider interval, reflecting greater certainty that the true parameter falls within the range.
How It's Calculated
For a population mean (when the population standard deviation is unknown, which is common), the confidence interval is typically calculated using the t-distribution. The formula is:
CI = ׯ ± t * (s / √n)
Where:
ׯ (x-bar) is the Sample Mean.
s is the Sample Standard Deviation.
n is the Sample Size.
t is the t-critical value. This value depends on the confidence level and the degrees of freedom (which is n - 1 for a one-sample mean). The t-critical value is found using statistical tables or software and represents how many standard errors away from the mean the interval's endpoints will be.
s / √n is the Standard Error of the Mean (SEM).
The Calculator's Logic
This calculator uses the following steps:
It takes your provided Sample Mean, Sample Standard Deviation, Sample Size, and desired Confidence Level.
It calculates the Degrees of Freedom (df) as n - 1.
It determines the t-critical value based on the confidence level and degrees of freedom. Since standard JavaScript doesn't have a built-in inverse t-distribution function, this calculator uses a common approximation or a simplified approach for illustrative purposes (in a real-world, highly precise application, a statistical library would be preferred). For simplicity and common use cases with larger sample sizes, we'll often approximate the t-value using the z-score from the standard normal distribution, especially for confidence levels like 90%, 95%, 99%. For this calculator, we will use approximations for common confidence levels.
It calculates the Margin of Error: Margin of Error = t * (s / √n).
Finally, it computes the Confidence Interval:
Lower Bound: ׯ - Margin of Error
Upper Bound: ׯ + Margin of Error
Note: Calculating the exact t-critical value requires advanced statistical functions not natively available in basic JavaScript. This calculator uses common approximations for the t-values corresponding to 90%, 95%, and 99% confidence levels for moderate to large sample sizes. For high precision or small sample sizes, a dedicated statistical software or library is recommended.
Use Cases
Confidence intervals are widely used in:
Market Research: Estimating the average spending of customers within a certain range.
Medical Studies: Determining the likely range for the effectiveness of a new drug or treatment.
Quality Control: Setting acceptable ranges for product measurements.
Social Sciences: Estimating average opinions or behaviors in a population.
A/B Testing: Understanding the range of potential improvement from a website change.
By providing a range instead of a single point estimate, confidence intervals offer a more realistic picture of the uncertainty inherent in statistical inference.
function calculateConfidenceInterval() {
var sampleMean = parseFloat(document.getElementById("sampleMean").value);
var sampleStdDev = parseFloat(document.getElementById("sampleStdDev").value);
var sampleSize = parseInt(document.getElementById("sampleSize").value, 10);
var confidenceLevel = parseFloat(document.getElementById("confidenceLevel").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
resultDiv.style.display = 'none';
// Input validation
if (isNaN(sampleMean) || isNaN(sampleStdDev) || isNaN(sampleSize) || isNaN(confidenceLevel)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all fields.";
resultDiv.style.backgroundColor = '#dc3545'; // Red for error
resultDiv.style.display = 'block';
return;
}
if (sampleStdDev <= 0) {
resultDiv.innerHTML = "Error: Sample Standard Deviation must be positive.";
resultDiv.style.backgroundColor = '#dc3545'; // Red for error
resultDiv.style.display = 'block';
return;
}
if (sampleSize <= 1) {
resultDiv.innerHTML = "Error: Sample Size must be greater than 1.";
resultDiv.style.backgroundColor = '#dc3545'; // Red for error
resultDiv.style.display = 'block';
return;
}
if (confidenceLevel = 1) {
resultDiv.innerHTML = "Error: Confidence Level must be between 0 and 1 (e.g., 0.95).";
resultDiv.style.backgroundColor = '#dc3545'; // Red for error
resultDiv.style.display = 'block';
return;
}
// Approximate t-critical values for common confidence levels and reasonably large n
// In a real application, use a proper inverse t-distribution function or lookup table.
var alpha = 1 – confidenceLevel;
var alphaOver2 = alpha / 2;
var t_critical;
// Approximations for common confidence levels (using z-scores as approximation for larger n)
// For more accuracy, especially with small n, a lookup or library is needed.
if (Math.abs(confidenceLevel – 0.90) t is higher for small n, e.g., t(0.05, 29) = 1.699
// 95%: 1.960 (z) -> t is higher for small n, e.g., t(0.025, 29) = 2.045
// 99%: 2.576 (z) -> t is higher for small n, e.g., t(0.005, 29) = 2.756
if (sampleSize > 30) { // Approximation using Z-scores for larger samples
t_critical = 1.645;
} else if (sampleSize > 10) {
t_critical = 1.812; // Approximation for moderate n
} else {
t_critical = 2.262; // Approximation for small n (e.g., df=9)
}
} else if (Math.abs(confidenceLevel – 0.95) 30) {
t_critical = 1.960;
} else if (sampleSize > 10) {
t_critical = 2.228; // Approximation for moderate n
} else {
t_critical = 3.169; // Approximation for small n (e.g., df=9)
}
} else if (Math.abs(confidenceLevel – 0.99) 30) {
t_critical = 2.576;
} else if (sampleSize > 10) {
t_critical = 2.763; // Approximation for moderate n
} else {
t_critical = 4.140; // Approximation for small n (e.g., df=9)
}
} else {
// For other confidence levels, a more robust method is needed.
// Using Z-score as a fallback if confidence level isn't standard (less accurate for small n)
// This part is a significant simplification.
var z_map = {
0.80: 1.282, 0.85: 1.440, 0.90: 1.645, 0.95: 1.960, 0.99: 2.576, 0.995: 2.807
};
t_critical = z_map[confidenceLevel] || 1.960; // Default to 95% Z if not found
resultDiv.innerHTML = "Warning: Using Z-score approximation for non-standard confidence level or small sample size. Results may be less accurate.";
resultDiv.style.backgroundColor = '#ffc107'; // Yellow for warning
resultDiv.style.display = 'block';
}
var standardError = sampleStdDev / Math.sqrt(sampleSize);
var marginOfError = t_critical * standardError;
var lowerBound = sampleMean – marginOfError;
var upperBound = sampleMean + marginOfError;
resultDiv.innerHTML = "Confidence Interval (" + (confidenceLevel * 100) + "%): [" + lowerBound.toFixed(4) + ", " + upperBound.toFixed(4) + "]";
resultDiv.style.backgroundColor = 'var(–success-green)'; // Green for success
resultDiv.style.display = 'block';
}