How to Calculate 95 Confidence Interval

95% Confidence Interval Calculator

Use this calculator to determine the 95% confidence interval for a population mean, given your sample's mean, standard deviation, and size.

Sample Mean (x̄): Sample Standard Deviation (s): Sample Size (n):

function calculateConfidenceInterval() { var sampleMean = parseFloat(document.getElementById('sampleMean').value); var sampleStdDev = parseFloat(document.getElementById('sampleStdDev').value); var sampleSize = parseInt(document.getElementById('sampleSize').value); var resultDiv = document.getElementById('result'); // Validate inputs if (isNaN(sampleMean) || isNaN(sampleStdDev) || isNaN(sampleSize)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; return; } if (sampleSize <= 1) { resultDiv.innerHTML = 'Sample Size must be greater than 1.'; return; } if (sampleStdDev 30) // For smaller samples, a t-distribution would be more appropriate, // but for simplicity and common use, we'll use the Z-score here. var zScore = 1.96; // Calculate Standard Error var standardError = sampleStdDev / Math.sqrt(sampleSize); // Calculate Margin of Error var marginOfError = zScore * standardError; // Calculate Confidence Interval var lowerBound = sampleMean – marginOfError; var upperBound = sampleMean + marginOfError; resultDiv.innerHTML = '

Calculation Results:

' + 'Sample Mean (x̄): ' + sampleMean.toFixed(2) + " + 'Sample Standard Deviation (s): ' + sampleStdDev.toFixed(2) + " + 'Sample Size (n): ' + sampleSize + " + 'Z-score (for 95% CI): ' + zScore.toFixed(2) + " + 'Standard Error: ' + standardError.toFixed(4) + " + 'Margin of Error: ' + marginOfError.toFixed(4) + " + 'The 95% Confidence Interval is: [' + lowerBound.toFixed(2) + ', ' + upperBound.toFixed(2) + ']' + 'This means we are 95% confident that the true population mean lies between ' + lowerBound.toFixed(2) + ' and ' + upperBound.toFixed(2) + '.'; } .confidence-interval-calculator { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 700px; margin: 30px auto; border: 1px solid #e0e0e0; } .confidence-interval-calculator h2 { color: #2c3e50; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .confidence-interval-calculator p { color: #34495e; line-height: 1.6; margin-bottom: 15px; } .calculator-form label { display: block; margin-bottom: 8px; color: #34495e; font-weight: bold; } .calculator-form input[type="number"] { width: calc(100% – 22px); padding: 12px; margin-bottom: 18px; border: 1px solid #ccc; border-radius: 6px; font-size: 1em; box-sizing: border-box; transition: border-color 0.3s ease; } .calculator-form input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.3); } .calculator-form button { background-color: #007bff; color: white; padding: 13px 25px; border: none; border-radius: 6px; cursor: pointer; font-size: 1.1em; width: 100%; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 10px; } .calculator-form button:hover { background-color: #0056b3; transform: translateY(-2px); } .calculator-form button:active { transform: translateY(0); } .calculator-result { background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 8px; padding: 20px; margin-top: 25px; color: #155724; font-size: 1.05em; line-height: 1.6; } .calculator-result h3 { color: #0f5132; margin-top: 0; margin-bottom: 15px; font-size: 1.5em; } .calculator-result p { margin-bottom: 10px; } .calculator-result p strong { color: #0f5132; }

Understanding the 95% Confidence Interval

What is a Confidence Interval?

In statistics, a confidence interval (CI) is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It provides a way to express the precision and uncertainty associated with a sample estimate. Instead of providing a single point estimate (like the sample mean), a confidence interval gives a range, along with a confidence level, indicating how likely it is that the interval contains the true population parameter.

What Does "95% Confidence" Mean?

A 95% confidence interval means that if you were to take many random samples from the same population and construct a confidence interval for each sample, approximately 95% of those intervals would contain the true population parameter (e.g., the true population mean). It does NOT mean there is a 95% probability that the true population mean falls within a specific calculated interval. Once an interval is calculated, the true mean either is or isn't within it; the 95% refers to the reliability of the estimation method over many repetitions.

Why is the 95% Confidence Interval Important?

Quantifies Uncertainty: It provides a measure of the uncertainty around a sample estimate. A wider interval indicates more uncertainty, while a narrower interval suggests greater precision.
Inference about Population: It allows researchers to make inferences about a population parameter based on a sample, which is often more practical than studying an entire population.
Decision Making: In fields like medicine, engineering, and business, confidence intervals help in making informed decisions by understanding the potential range of outcomes.
Hypothesis Testing: Confidence intervals are closely related to hypothesis testing. If a hypothesized population mean falls outside the 95% CI, it suggests that the hypothesis might be incorrect at the 0.05 significance level.

How to Calculate a 95% Confidence Interval for the Mean

The general formula for a confidence interval for the population mean (when the population standard deviation is unknown and the sample size is sufficiently large, typically n > 30) is:

CI = Sample Mean ± (Z-score * (Sample Standard Deviation / sqrt(Sample Size)))

Let's break down the components:

Sample Mean (x̄): This is the average of your sample data. It's your best point estimate for the true population mean.
Z-score: This value corresponds to your chosen confidence level. For a 95% confidence interval, the Z-score is typically 1.96. This value comes from the standard normal distribution and represents how many standard deviations away from the mean you need to go to capture 95% of the data.
Sample Standard Deviation (s): This measures the amount of variation or dispersion of your sample data points around the sample mean.
Sample Size (n): The number of observations in your sample.
Standard Error (SE): Calculated as Sample Standard Deviation / sqrt(Sample Size). It estimates the standard deviation of the sampling distribution of the sample mean.
Margin of Error (ME): Calculated as Z-score * Standard Error. This is the amount added to and subtracted from the sample mean to create the interval.

Interpreting the Results

When you calculate a 95% confidence interval, say [45.2, 54.8], you would state: "We are 95% confident that the true population mean lies between 45.2 and 54.8." This means that if we were to repeat our sampling process many times, 95% of the confidence intervals constructed would contain the true population mean.

Limitations

Sample Quality: The validity of the confidence interval heavily relies on the sample being representative of the population. Biased samples will lead to misleading intervals.
Assumptions: The calculation often assumes that the data is normally distributed or that the sample size is large enough for the Central Limit Theorem to apply.
Not a Probability for a Single Interval: It's crucial to remember that the 95% refers to the method's reliability, not the probability that a specific, already calculated interval contains the true mean.

How to Use the Calculator

To use the 95% Confidence Interval Calculator, simply input the following values from your sample data:

Sample Mean (x̄): The average value of your observations.
Sample Standard Deviation (s): The measure of spread in your sample.
Sample Size (n): The total number of data points in your sample.

Click "Calculate 95% CI," and the calculator will instantly provide the lower and upper bounds of your 95% confidence interval, along with the intermediate steps.

Examples

Example 1: Average Test Scores

A teacher wants to estimate the average test score of all students in a large district. They take a random sample of 50 students and find:

Sample Mean (x̄): 75 points
Sample Standard Deviation (s): 12 points
Sample Size (n): 50 students

Using the calculator:

Z-score (95% CI) = 1.96
Standard Error = 12 / sqrt(50) ≈ 1.697
Margin of Error = 1.96 * 1.697 ≈ 3.326
Lower Bound = 75 – 3.326 = 71.674
Upper Bound = 75 + 3.326 = 78.326

The 95% Confidence Interval is approximately [71.67, 78.33]. The teacher can be 95% confident that the true average test score for all students in the district lies between 71.67 and 78.33 points.

Example 2: Product Lifespan

A manufacturer tests a sample of 100 light bulbs to estimate the average lifespan of a new product line:

Sample Mean (x̄): 1200 hours
Sample Standard Deviation (s): 150 hours
Sample Size (n): 100 bulbs