Confidence Interval Calculator

Estimate Population Values with Confidence

In statistics, a confidence interval provides a range of values which is likely to contain the true population mean. It's a way of quantifying the uncertainty of a sample estimate. This calculator computes the confidence interval for a sample mean at various common confidence levels.

The Formula Explained

The formula for a confidence interval is: CI = x̄ ± Z * (s / √n)

• x̄: The sample mean (the average of your sample data).
• Z: The Z-score, a value determined by your desired confidence level. A higher confidence level requires a higher Z-score.
• s: The sample standard deviation.
• n: The sample size.
• (s / √n): This part is known as the standard error.

How to Use the Calculator

1. Enter Sample Data: Input your sample mean, sample standard deviation, and sample size.
2. Select Confidence Level: Choose your desired confidence level (e.g., 95%, 99%). 95% is the most common.
3. Calculate: The tool will display the lower and upper bounds of the confidence interval.

Real-World Example

A researcher tests the IQ of a sample of 40 students. The sample has a mean IQ of 105 and a standard deviation of 15. The researcher wants to find the 95% confidence interval for the true mean IQ of all students.

• Sample Mean (x̄): 105
• Standard Deviation (s): 15
• Sample Size (n): 40
• Confidence Level: 95% (which corresponds to a Z-score of 1.96)
• Standard Error: 15 / √40 ≈ 2.37
• Margin of Error: 1.96 * 2.37 ≈ 4.65
• Confidence Interval: 105 ± 4.65, which is (100.35, 109.65). We can be 95% confident that the true mean IQ of the entire student population lies between 100.35 and 109.65.

Frequently Asked Questions (FAQ)

• What does a 95% confidence level mean? It means that if we were to take many samples and build a confidence interval from each one, we would expect about 95% of those intervals to contain the true population mean.
• How does sample size affect the confidence interval? A larger sample size (n) will result in a smaller standard error and therefore a narrower, more precise confidence interval.
• What is a Z-score? A Z-score represents how many standard deviations a value is from the mean in a standard normal distribution. For confidence intervals, it defines the boundaries that capture a certain percentage of the probability.