Confidence interval calculator

Confidence Interval Calculator — Accurate Confidence Range Estimator

Use this free Confidence Interval Calculator to find how confident you can be that your sample mean reflects the true population mean. The tool supports both Z-distribution (when population standard deviation σ is known) and t-distribution (when σ is unknown), letting you compute confidence intervals for 90%, 95%, and 99% levels.

What is a Confidence Interval?

A confidence interval (CI) is a range of values likely to contain the true population parameter. For example, a 95% CI of [48.3, 51.7] means we are 95% confident that the true population mean lies between 48.3 and 51.7. This interval reflects both the sample variability and the confidence level.

Confidence Interval Formulas

1. Z-based (σ known): CI = x̄ ± Z × (σ / √n)

2. t-based (σ unknown): CI = x̄ ± t × (s / √n)

Where:

• x̄ = sample mean
• σ = population standard deviation (if known)
• s = sample standard deviation (if σ unknown)
• n = sample size
• Z = Z critical value (from standard normal distribution)
• t = t critical value (depends on degrees of freedom = n − 1)

How to Use This Calculator

1. Enter your sample mean, standard deviation (σ or s), and sample size (n).
2. Select the confidence level (90%, 95%, or 99%).
3. Choose the distribution type — Z (known σ) or t (unknown σ).
4. Click Calculate to view the margin of error and full confidence interval.
5. Review the interpretation guide below to understand your results.

Example Calculation

Suppose your sample mean is 50, sample standard deviation is 4, and n = 25, with a 95% confidence level (t = 2.064). The standard error = 4 / √25 = 0.8. Margin of error = 2.064 × 0.8 = 1.651. So, the confidence interval is: 50 ± 1.651 → [48.35, 51.65]. We can be 95% confident that the true population mean lies between 48.35 and 51.65.

Interpretation of Confidence Levels

• 90% — smaller interval, less confidence, more precision.
• 95% — standard choice for most studies and research.
• 99% — wider interval, more confidence but less precision.

When to Use Z vs. t Distribution

• Z-distribution: use if population standard deviation (σ) is known and n ≥ 30.
• t-distribution: use if σ is unknown or n < 30 (small sample).

Related Concepts

• Standard Error (SE) — measures variability of sample mean.
• Central Limit Theorem — explains why normal distribution applies.
• Confidence Interval — Wikipedia
• Statology — Confidence Interval Formula Explained