Confidence Interval Calculator

Confidence Interval Calculator

Calculate confidence intervals for a population mean using either the z-distribution (when population standard deviation is known) or the t-distribution (when using sample standard deviation).

Sample Mean (x̄)

Standard Deviation (σ or s)

Sample Size (n)

Confidence Level90%95%99%

Standard Deviation TypePopulation (σ) - Use z-distributionSample (s) - Use t-distribution

Calculate Confidence IntervalClear

How to Use This Calculator

1. Enter the sample mean (x̄): The average value from your sample
2. Enter the standard deviation: Either population (σ) or sample (s) standard deviation
3. Enter the sample size (n): Number of observations in your sample
4. Select confidence level: Choose 90%, 95%, or 99%
5. Select standard deviation type: Choose whether you’re using population or sample standard deviation
6. Click Calculate: Get your confidence interval with full explanation

Understanding Confidence Intervals

A confidence interval provides a range of plausible values for an unknown population parameter. Instead of a single point estimate, you get:

• Lower bound: The lowest plausible value
• Upper bound: The highest plausible value
• Margin of error: How much uncertainty exists in your estimate
• Confidence level: The probability that your method captures the true parameter

Example Interpretation

If a 95% confidence interval for average height is (165 cm, 175 cm):

✅ Correct: “We are 95% confident that the true population mean height is between 165 cm and 175 cm.”

❌ Incorrect: “There is a 95% probability that the mean is in this interval.” (The parameter is fixed, not random!)

When to Use z vs t Distribution

Use z-distribution when:

• Population standard deviation (σ) is known
• Any sample size

Use t-distribution when:

• Population standard deviation is unknown (using sample s)
• Especially important for small samples (n < 30)
• As sample size increases, t-distribution approaches z-distribution

Formula

The general form of a confidence interval for a mean is:

$\text{CI} = \bar{x} \pm (\text{critical value}) \times \frac{\sigma}{\sqrt{n}}$

Where:

• $\bar{x}$ = sample mean
• $\sigma$ = standard deviation
• $n$ = sample size
• Critical value = $z$ or $t$ value depending on the situation

Factors Affecting Interval Width

Three factors influence how wide your confidence interval is:

1. Confidence Level

• Higher confidence → Wider interval
• 99% CI is wider than 95% CI

2. Sample Size

• Larger sample → Narrower interval
• Standard error decreases as √n increases

3. Variability

• More spread in data → Wider interval
• Higher standard deviation = more uncertainty

Common Applications

• Clinical trials: Estimate treatment effects
• Quality control: Monitor production processes
• Market research: Estimate customer satisfaction scores
• Education: Assess average test performance
• Finance: Estimate average returns

Related Resources

• Confidence Intervals Lesson - Learn the theory
• T-Test Calculator - Compare means
• Sample Size Calculator - Plan your study
• T-Table - Critical t-values reference

Frequently Asked Questions

What is a confidence interval?

A confidence interval is a range of values that is likely to contain the true population parameter. A 95% confidence interval means that if you repeated the study 100 times, approximately 95 of those intervals would contain the true population mean.

How do I calculate a 95% confidence interval?

Use this formula: CI = x̄ ± (critical value × standard error). For 95% confidence with known σ, use z = 1.96. For unknown σ, use the t-critical value for your degrees of freedom. Enter your values in the calculator above for instant results.

What is the margin of error in a confidence interval?

The margin of error is half the width of the confidence interval. It equals (critical value) × (standard error). A smaller margin of error means more precise estimation but requires larger sample sizes or less variable data.

What confidence level should I use?

The most common choice is 95%, which balances precision and confidence. Use 99% when errors are costly (medical studies), or 90% when you need a narrower interval and can tolerate more uncertainty.

Why is my confidence interval so wide?

Wide confidence intervals result from: small sample sizes, high data variability, or high confidence levels (99% vs 95%). To narrow your interval, increase sample size, reduce measurement error, or accept a lower confidence level.

What’s the difference between z and t intervals?

Use z-intervals when the population standard deviation (σ) is known (rare in practice). Use t-intervals when using sample standard deviation (s), which is almost always the case. For large samples (n > 30), they give similar results.