Confidence Interval Calculator
Calculate the confidence interval for a population mean. Choose between using the z-statistic (when population standard deviation is known) or the t-statistic (when population standard deviation is unknown). This calculator provides the interval, margin of error, and a detailed step-by-step breakdown of the calculation.
What Is a Confidence Interval?
A confidence interval is a range of values that is likely to contain the true population parameter (such as the mean) with a certain level of confidence. It provides a measure of the uncertainty associated with a sample estimate.
Formula: x̄ ± (critical value) × (standard error)
where standard error = s / √n (or σ / √n if known)
Choosing z vs. t
- z-statistic: Use when the population standard deviation (σ) is known.
- t-statistic: Use when the population standard deviation (σ) is unknown and estimated by the sample standard deviation (s).
The calculator automatically determines which statistic to use based on whether you provide σ. If σ is not provided, it uses the t-distribution with (n − 1) degrees of freedom.
How the Calculator Works
- Compute Standard Error: SE = s / √n (or σ / √n if known).
- Find Critical Value: z* from the normal distribution or t* from the t-distribution.
- Calculate Margin of Error: ME = critical value × standard error.
- Determine Interval: Lower = x̄ − ME, Upper = x̄ + ME.
Why Use This Confidence Interval Calculator?
- z or t: Automatically selects the appropriate method.
- Step-by-Step: See the calculation process.
- Visualization: Understand the interval graphically.
- Free & Private: No registration, no data storage.
❓ Confidence Interval Calculator FAQ
What is a confidence interval?
A confidence interval is a range of values that is likely to contain the true population parameter (e.g., the mean) with a certain level of confidence.
What is the difference between z and t confidence intervals?
A z-interval is used when the population standard deviation is known. A t-interval is used when the population standard deviation is unknown and estimated from the sample.
What is the margin of error?
The margin of error is the maximum expected difference between the sample estimate and the true population parameter. It is calculated as critical value × standard error.
How do I choose the confidence level?
Common confidence levels are 90%, 95%, and 99%. A higher confidence level produces a wider interval (more certain but less precise).
What is the standard error?
The standard error measures the variability of the sample mean. It is calculated as s / √n (or σ / √n if known).
When should I use the t-distribution?
Use the t-distribution when the population standard deviation is unknown and the sample size is small (typically n < 30) or when the sample is drawn from a normally distributed population.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.