How to Calculate the Confidence Interval

Confidence Interval Calculator

function calculateConfidenceInterval() { var sampleMean = parseFloat(document.getElementById('sampleMean').value); var sampleStdDev = parseFloat(document.getElementById('sampleStdDev').value); var sampleSize = parseInt(document.getElementById('sampleSize').value); var confidenceLevel = parseFloat(document.getElementById('confidenceLevel').value); var resultDiv = document.getElementById('confidenceIntervalResult'); if (isNaN(sampleMean) || isNaN(sampleStdDev) || isNaN(sampleSize) || isNaN(confidenceLevel)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; return; } if (sampleStdDev < 0) { resultDiv.innerHTML = 'Sample Standard Deviation cannot be negative.'; return; } if (sampleSize <= 1) { resultDiv.innerHTML = 'Sample Size must be greater than 1.'; return; } var zScore; switch (confidenceLevel) { case 90: zScore = 1.645; break; case 95: zScore = 1.96; break; case 99: zScore = 2.576; break; default: resultDiv.innerHTML = 'Invalid Confidence Level selected.'; return; } var standardError = sampleStdDev / Math.sqrt(sampleSize); var marginOfError = zScore * standardError; var lowerBound = sampleMean – marginOfError; var upperBound = sampleMean + marginOfError; resultDiv.innerHTML = 'With ' + confidenceLevel + '% confidence, the true population mean lies between:' + '[' + lowerBound.toFixed(4) + ', ' + upperBound.toFixed(4) + ']' + 'Margin of Error: ' + marginOfError.toFixed(4) + ''; } // Initial calculation on page load for default values window.onload = calculateConfidenceInterval;

Understanding Confidence Intervals

In statistics, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data, that might contain the true value of an unobserved population parameter. It expresses the range within which we expect the true population parameter (like the mean) to lie, with a certain level of confidence.

What Does a Confidence Interval Tell Us?

When you calculate a 95% confidence interval, it means that if you were to take many random samples from the same population and construct a confidence interval from each sample, approximately 95% of those intervals would contain the true population parameter. It does NOT mean there is a 95% probability that the true mean falls within your specific calculated interval.

Components of a Confidence Interval

The calculation of a confidence interval for a population mean typically involves several key components:

• Sample Mean (x̄): This is the average value of your sample data. It's your best single estimate of the population mean.
• Sample Standard Deviation (s): This measures the amount of variation or dispersion of your sample data. A larger standard deviation indicates more spread-out data.
• Sample Size (n): The number of observations in your sample. Larger sample sizes generally lead to narrower confidence intervals, as they provide more information about the population.
• Confidence Level: This is the probability that the interval estimate will contain the population parameter. Common confidence levels are 90%, 95%, and 99%.
• Critical Value (Z-score or t-score): This value is determined by your chosen confidence level and, for smaller sample sizes, the degrees of freedom (related to sample size). For larger sample sizes (typically n > 30), a Z-score is often used.
• Standard Error (SE): This is the standard deviation of the sampling distribution of the sample mean. It's calculated as s / sqrt(n).
• Margin of Error (ME): This is the range above and below the sample mean that defines the confidence interval. It's calculated as Critical Value * Standard Error.

How to Interpret a Confidence Interval

Let's say you calculate a 95% confidence interval for the average height of students in a university to be [165 cm, 175 cm]. This means you are 95% confident that the true average height of ALL students in that university falls somewhere between 165 cm and 175 cm.

A wider interval indicates more uncertainty, while a narrower interval indicates more precision in your estimate. The width is influenced by the sample size, standard deviation, and the chosen confidence level.

Factors Affecting the Width of a Confidence Interval

• Sample Size: Increasing the sample size (n) decreases the standard error, leading to a narrower confidence interval. More data generally means more precision.
• Standard Deviation: A smaller sample standard deviation (s) indicates less variability in the data, resulting in a narrower confidence interval.
• Confidence Level: A higher confidence level (e.g., 99% vs. 95%) requires a larger critical value, which in turn leads to a wider confidence interval. To be more confident that your interval captures the true parameter, you need to make the interval wider.

Example Calculation

Imagine a researcher wants to estimate the average score on a new standardized test. They take a random sample of 100 students and find the following:

• Sample Mean (x̄) = 75
• Sample Standard Deviation (s) = 12
• Sample Size (n) = 100
• Desired Confidence Level = 95%

Using the calculator above:

1. Critical Value (Z-score): For a 95% confidence level, the Z-score is 1.96.
2. Standard Error (SE): SE = s / sqrt(n) = 12 / sqrt(100) = 12 / 10 = 1.2
3. Margin of Error (ME): ME = Z-score * SE = 1.96 * 1.2 = 2.352
4. Confidence Interval:
   • Lower Bound = Sample Mean - ME = 75 - 2.352 = 72.648
   • Upper Bound = Sample Mean + ME = 75 + 2.352 = 77.352

So, the 95% confidence interval for the average test score is [72.648, 77.352]. This means the researcher is 95% confident that the true average score for all students on this test lies between 72.648 and 77.352.