Confidence Interval Calculator
Calculate confidence intervals, margin of error, and standard error for population means from sample statistics.
95% Confidence Interval
(46.4215, 53.5785)
Margin of Error
3.5785
Standard Error
1.8257
CI = x̄ ± z × (σ / sqrt(n))How to Use the Confidence Interval Calculator
- Select your desired confidence level: 90%, 95%, or 99%.
- Enter the sample mean (x̄), standard deviation (σ or s), and sample size (n).
- Review the confidence interval range, margin of error (E), and standard error (SE).
Formulas & Principles
Standard Error: SE = σ / √n
Margin of Error: ME = Z* × (σ / √n)
Confidence Interval: CI = x̄ ± ME = (x̄ - ME, x̄ + ME)
Critical Z Values: 90% → Z = 1.645 | 95% → Z = 1.960 | 99% → Z = 2.576
Frequently Asked Questions
What does a 95% confidence interval mean?
It means that if you took repeated independent samples from the same population and calculated a 95% confidence interval for each, 95% of those calculated intervals would contain the true population parameter.
How does increasing sample size affect the confidence interval?
Increasing the sample size (n) reduces the standard error (dividing by √n), which shrinks the margin of error and creates a narrower, more precise interval.
When should I use a Z-interval versus a T-interval?
Use a Z-interval when the population standard deviation is known or the sample size is large (n ≥ 30) by the Central Limit Theorem. Use a T-interval with unknown variance for small samples.
Why this calculator matters
This calculator helps solve mathematical problems faster and with fewer manual arithmetic mistakes. It is useful for studying, research verification, technical engineering checks, and everyday problem-solving.
When to use it
Use it when you need a fast answer, a cross-check for manual calculations, or a clear step-by-step verification of your result.
Quick tips
Double-check your inputs, keep units consistent across all fields, and verify boundary conditions.
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