What is the formula for the confidence interval of a mean?

The formula is `CI = x̄ ± z* × (s / √n)`, where `x̄` is sample mean, `z*` is critical value, `s` is standard deviation, and `n` is sample size.

When should I use a t-distribution instead of a z-distribution?

Use a t-distribution when the population standard deviation is unknown and the sample size is small ($n < 30$).

What are common Z critical values?

For 90% confidence, `Z* = 1.645`; for 95% confidence, `Z* = 1.960`; for 99% confidence, `Z* = 2.576`.

How does increasing sample size affect the confidence interval width?

Increasing sample size ($n$) decreases the standard error ($s / \sqrt{n}$), making the margin of error smaller and the confidence interval narrower and more precise.

What is the margin of error?

The margin of error is the radius of the confidence interval: `ME = z* × (s / √n)`.

What Z-score values correspond to standard 90%, 95%, and 99% confidence levels?

A 90% confidence level uses `Z = 1.645`, 95% uses `Z = 1.960`, and 99% uses `Z = 2.576` under a normal Gaussian distribution.

Math & Statistics Cluster

Confidence Interval Calculator (90%, 95%, 99% Confidence Limits)

Calculate confidence intervals for sample means and proportions with margin of error.

TL;DR: Confidence Interval Calculator computes the upper and lower confidence bounds for a population parameter: $\text{CI} = \bar{x} \pm Z^* \times (s / \sqrt{n})$.

Sample mean

Standard deviation

Sample size (n)

Confidence level

What Is a Confidence Interval and What Does a 95% Confidence Level Mean?

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter. A 95% confidence level means that if the same study were repeated 100 times with new random samples, 95 of the 100 resulting intervals would contain the true population mean.

How to Use the Confidence Interval Calculator

Our Confidence Interval Calculator performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.

Select parameter type: Mean (Average) OR Proportion (Percentage).
Enter the Sample Mean ($\bar{x}$), Sample Size ($n$), and Standard Deviation ($s$).
Select desired Confidence Level (90%, 95%, 99%).
Choose Z-score ($n \ge 30$) or Student's t-distribution ($n < 30$).
View calculated Margin of Error ($ME$), Lower Bound, and Upper Bound.

Mathematical Formula & Equations

Calculates margin of error and confidence intervals (90%, 95%, 99%) for sample means and survey proportions.

\[ \text{CI} = \bar{x} \pm Z^* \frac{s}{\sqrt{n}} \]

Calculation Example

For sample mean 100, $s = 15$, $n = 100$ at 95% confidence ($Z^* = 1.96$): Margin of error = $1.96 \times 1.5 = 2.94$. 95% CI is $[97.06, 102.94]$.

100% Client-Side Privacy & Data Security

All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.

Frequently Asked Questions

: The formula is `CI = x̄ ± z* × (s / √n)`, where `x̄` is sample mean, `z*` is critical value, `s` is standard deviation, and `n` is sample size.
: Use a t-distribution when the population standard deviation is unknown and the sample size is small ($n < 30$).
: For 90% confidence, `Z* = 1.645`; for 95% confidence, `Z* = 1.960`; for 99% confidence, `Z* = 2.576`.
: Increasing sample size ($n$) decreases the standard error ($s / \sqrt{n}$), making the margin of error smaller and the confidence interval narrower and more precise.
: The margin of error is the radius of the confidence interval: `ME = z* × (s / √n)`.
: A 90% confidence level uses `Z = 1.645`, 95% uses `Z = 1.960`, and 99% uses `Z = 2.576` under a normal Gaussian distribution.

Cite This Calculator

Need to reference this online calculator in a financial proposal, academic paper, blog post, or classroom curriculum? Copy a citation below.

NexLove.org. (2026). Confidence Interval Calculator [Online Calculation Tool]. https://nexlove.org/tools/confidence-interval-calculator

“Confidence Interval Calculator.” NexLove.org, 2026, nexlove.org/tools/confidence-interval-calculator.

NexLove.org (2026) Confidence Interval Calculator. Available at: https://nexlove.org/tools/confidence-interval-calculator (Accessed: 2026).

@misc{nexlove_confidence_interval_calculator,
  title = {Confidence Interval Calculator},
  author = {{NexLove.org}},
  year = {2026},
  url = {https://nexlove.org/tools/confidence-interval-calculator}
}

Embed This Tool

Add the live Confidence Interval Calculator to your own website with this snippet — it loads the real, working calculator in an iframe.

<iframe src="https://nexlove.org/embed/confidence-interval-calculator.html" width="100%" height="600" style="border:1px solid #e2e8f0;border-radius:12px" title="Confidence Interval Calculator — NexLove.org" loading="lazy"></iframe>

Related Math & Statistics

Log & Exponent Calculator

Sample Size Calculator

Browse all Calculators