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Confidence
Interval.

The confidence interval calculator computes the confidence interval of the mean (using the normal or t-distribution) or of the standard deviation (using the chi-squared distribution). Results are reported in APA style with a full step-by-step breakdown.

What is a confidence interval?

The confidence interval is the range in which the population parameter is most likely to be found. The degree of certainty for which it is likely to be within that range is called the confidence level. When you collect sample data, you can not know the exact value of the parameter — only estimate it.

Mean confidence interval formula

When you know the population standard deviation σ (normal dist):

x̄ ± Z_α/2 · σ / √n

When you don't know σ, use the sample SD with the t distribution:

x̄ ± T_α/2(df) · S / √n

Standard deviation confidence interval formula

The statistic (n − 1)·S² / σ² follows a χ² distribution with n − 1 degrees of freedom.

(n − 1)·S² / χ²_1−α/2(df)  ≤  σ²  ≤  (n − 1)·S² / χ²_α/2(df)

// Confidence Interval

—

Fill in the form below.

Lower bound —

Upper bound —

Margin of Error —

Standard Error —

// Results — APA Style —

Confidence interval—

Margin of Error (MOE)—

Standard Error (SE)—

Alternative form—

Fill in the form to compute the confidence interval.

// Step by step

Steps will appear here once you enter your data.

// Calculation glossary

Confidence level (CL)—The certainty that the true parameter is inside the interval.
Mean / Average (x̄)—The sample average.
Margin of Error (MOE)—Half the width of the confidence interval.
Sample size (n)—The number of observations.
Significance level (α)—The uncertainty level, α = 1 − CL.
T / Z-score—Critical value from the t or normal distribution at p = 1 − α/2.
Degrees of freedom (df)—df = n − 1.
Sample SD (S)—Standard deviation computed from the sample.

// R Code

sigma=11
avg=10
n=21
MOE = qt(1 - 0.05/2, n - 1) * sigma / sqrt(n)
c(avg - MOE, avg + MOE)

Frequently asked questions

What does Confidence Interval Calculator do?

95 % / 99 % CI for proportions and means. The whole thing runs in your browser tab, so it is ready the moment the page finishes loading.

Is Confidence Interval Calculator free to use?

Yes. Confidence Interval Calculator is completely free on ToolMunch — no trial, no premium tier holding back the main feature, and no daily or hourly cap on how many times you can use it.

Do I need to create an account or sign up?

No. There is no account, no email address, and no login. Open Confidence Interval Calculator, do the job, and take the result — nothing is gated behind a signup form.

Are the numbers I enter sent to a server?

No. Confidence Interval Calculator runs entirely inside your browser tab, so what you type never leaves your own device. Nothing is uploaded, which means there is no copy of it on a server to worry about.

Does Confidence Interval Calculator work on Mac, Windows, and phones?

Yes. Confidence Interval Calculator runs in any modern web browser on Windows, macOS, Linux, Chromebook, Android, and iOS. There is nothing to download or install.

Can I use it as many times as I want?

Yes. There is no queue, no per-hour quota, and no counter that runs out — run Confidence Interval Calculator as often as the work needs.

ConfidenceInterval.