Confidence Interval Calculator

Use this calculator to determine the confidence interval for a population mean or proportion based on your sample data.

Calculate for Mean Calculate for Proportion

Sample Mean (x̄):

Sample Standard Deviation (s):

Sample Size (n):

Sample Proportion (p̂, as decimal):

Sample Size (n):

Confidence Level (%): 90% 95% 99%

Results:

' + 'Confidence Level: ' + confidenceLevel + '%' + 'Margin of Error: ' + moe.toFixed(4) + " + 'Confidence Interval: (' + lowerBound.toFixed(4) + ', ' + upperBound.toFixed(4) + ')'; } else { // Calculate for Proportion var sampleProportion = parseFloat(document.getElementById('sampleProportion').value); var sampleSizeProportion = parseInt(document.getElementById('sampleSizeProportion').value); if (isNaN(sampleProportion) || isNaN(sampleSizeProportion) || sampleSizeProportion <= 1 || sampleProportion 1) { resultDiv.innerHTML = 'Please enter valid numbers for Sample Proportion (between 0 and 1) and Sample Size (must be > 1).'; return; } var standardErrorProportion = Math.sqrt((sampleProportion * (1 – sampleProportion)) / sampleSizeProportion); moe = zScore * standardErrorProportion; lowerBound = sampleProportion – moe; upperBound = sampleProportion + moe; resultDiv.innerHTML = '

Results:

' + 'Confidence Level: ' + confidenceLevel + '%' + 'Margin of Error: ' + moe.toFixed(4) + " + 'Confidence Interval: (' + lowerBound.toFixed(4) + ', ' + upperBound.toFixed(4) + ')'; } } .confidence-interval-calculator label { display: inline-block; width: 250px; margin-bottom: 5px; font-weight: bold; } .confidence-interval-calculator input[type="number"], .confidence-interval-calculator select { width: 150px; padding: 8px; margin-bottom: 10px; border: 1px solid #ccc; border-radius: 4px; } .confidence-interval-calculator button { background-color: #007bff; color: white; padding: 10px 15px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; } .confidence-interval-calculator button:hover { background-color: #0056b3; } .confidence-interval-calculator #ciResult { background-color: #f9f9f9; border: 1px solid #ddd; padding: 15px; border-radius: 5px; } .confidence-interval-calculator #ciResult h3 { margin-top: 0; color: #333; }

Understanding the Confidence Interval Calculator

A confidence interval is a range of values, derived from sample statistics, 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.

What is a Confidence Interval?

When you conduct a study or experiment, you typically collect data from a sample of the population, not the entire population. For example, if you want to know the average height of all adults in a country, it's impractical to measure everyone. Instead, you take a sample and calculate the average height of that sample (the sample mean).

However, the sample mean is just an estimate. If you took another sample, you'd likely get a slightly different mean. A confidence interval addresses this by providing a range within which the true population mean (or proportion) is likely to fall, with a certain level of confidence.

For instance, a "95% confidence interval" means that if you were to take many samples and construct a confidence interval from each, about 95% of those intervals would contain the true population parameter.

Key Components of a Confidence Interval

Sample Mean (x̄) or Sample Proportion (p̂): This is the central estimate from your sample data.
Sample Standard Deviation (s): For means, this measures the spread or variability of data points in your sample.
Sample Size (n): The number of observations in your sample. Larger sample sizes generally lead to narrower (more precise) confidence intervals.
Confidence Level: This is the probability that the interval will contain the true population parameter. Common levels are 90%, 95%, and 99%. A higher confidence level results in a wider interval.
Margin of Error (MOE): This is the "plus or minus" amount that defines the width of the confidence interval around the sample estimate. It's calculated using the standard error and a critical value (Z-score or t-score).

How the Calculator Works

This calculator allows you to compute confidence intervals for two common scenarios:

1. Confidence Interval for a Population Mean

This is used when you want to estimate the true average value of a characteristic in a population. The formula used is:

CI = x̄ ± Z * (s / √n)

x̄: Sample Mean
Z: Z-score (critical value) corresponding to your chosen confidence level. For simplicity and common practical applications, especially with larger sample sizes (n > 30), this calculator uses Z-scores as an approximation for t-scores.
s: Sample Standard Deviation
n: Sample Size

The term s / √n is known as the Standard Error of the Mean (SEM).

2. Confidence Interval for a Population Proportion

This is used when you want to estimate the true proportion or percentage of a characteristic in a population (e.g., the percentage of people who prefer a certain product). The formula used is:

CI = p̂ ± Z * √[(p̂(1-p̂))/n]

p̂: Sample Proportion (expressed as a decimal, e.g., 0.6 for 60%)
Z: Z-score (critical value) corresponding to your chosen confidence level.
n: Sample Size

The term √[(p̂(1-p̂))/n] is known as the Standard Error of the Proportion (SEP).

Interpreting Your Results

The calculator will output a lower bound and an upper bound. For example, if you calculate a 95% confidence interval for the average height of (165 cm, 175 cm), it means you are 95% confident that the true average height of the entire population falls between 165 cm and 175 cm.

It's important to remember that a confidence interval does not mean there's a 95% chance the true mean is within *this specific interval*. Instead, it means that if you repeated the sampling process many times, 95% of the intervals constructed would contain the true population mean.

Examples

Example 1: Confidence Interval for Mean (Student Test Scores)

A teacher wants to estimate the average test score of all students in a large district. They randomly select 100 students and find the following:

Sample Mean (x̄): 75 points
Sample Standard Deviation (s): 12 points
Sample Size (n): 100 students
Confidence Level: 95%

Using the calculator:

Select "Calculate for Mean".
Enter Sample Mean: 75
Enter Sample Standard Deviation: 12
Enter Sample Size: 100
Select Confidence Level: 95%

Calculation:

Z-score for 95% CI = 1.96
Standard Error of Mean (SEM) = 12 / √100 = 12 / 10 = 1.2
Margin of Error (MOE) = 1.96 * 1.2 = 2.352
Lower Bound = 75 – 2.352 = 72.648
Upper Bound = 75 + 2.352 = 77.352

Result: The 95% confidence interval for the average test score is (72.648, 77.352). This means the teacher is 95% confident that the true average test score for all students in the district is between 72.65 and 77.35 points.

Example 2: Confidence Interval for Proportion (Product Preference)

A company surveys 500 potential customers to determine the proportion who prefer their new product. They find that 300 customers prefer the new product.

Number of successes: 300
Sample Size (n): 500
Sample Proportion (p̂): 300 / 500 = 0.6
Confidence Level: 90%

Using the calculator:

Select "Calculate for Proportion".
Enter Sample Proportion: 0.6
Enter Sample Size: 500
Select Confidence Level: 90%

Calculation:

Z-score for 90% CI = 1.645
Standard Error of Proportion (SEP) = √[(0.6 * (1 – 0.6)) / 500] = √[(0.6 * 0.4) / 500] = √[0.24 / 500] = √0.00048 ≈ 0.0219089
Margin of Error (MOE) = 1.645 * 0.0219089 ≈ 0.03601
Lower Bound = 0.6 – 0.03601 = 0.56399
Upper Bound = 0.6 + 0.03601 = 0.63601

Result: The 90% confidence interval for the proportion of customers who prefer the new product is (0.5640, 0.6360). This means the company is 90% confident that between 56.4% and 63.6% of all potential customers prefer their new product.