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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.

// FILL DETAILS
// 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)

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 ± Zα/2 · σ / √n

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

x̄ ± Tα/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 / χ²1−α/2(df)  ≤  σ²  ≤  (n − 1)·S² / χ²α/2(df)