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📊 Confidence Interval Calculator

Calculate 90%, 95%, or 99% confidence intervals for a population mean (known σ or sample s) or a proportion. Enter your sample statistics and instantly get the lower bound, upper bound, margin of error, and an interpretation.

Confidence Interval Formulas

Mean CI (z-interval)
ME = z × σ / √n
CI = [ x̄ − ME, x̄ + ME ]
Proportion CI
SE = √( p̂(1 − p̂) / n )
ME = z × SE
CI = [ p̂ − ME, p̂ + ME ]

The z-value depends on your chosen confidence level: 90% → 1.645, 95% → 1.96, 99% → 2.576. A wider interval gives more confidence but less precision.

How to Calculate a Confidence Interval

  1. 1
    Choose the Type
    Select "Mean" if you are estimating a population mean using a known or sample standard deviation. Select "Proportion" for a success rate or percentage.
  2. 2
    Enter Your Sample Stats
    Provide your sample mean (or proportion), standard deviation, and sample size. For proportions you can type "42/200" and the calculator will parse it.
  3. 3
    Choose Confidence Level
    Common choices are 95% (science/business) and 99% (medical/safety). Higher confidence widens the interval.
  4. 4
    Read the Interval
    The result shows your lower bound, upper bound, margin of error, and a plain-English interpretation.

Worked Example

A sample of n = 40 students has a mean test score of x̄ = 72 with a standard deviation of σ = 10. Find the 95% CI.

z = 1.96 (for 95% CI)
ME = 1.96 × 10 / √40 = 1.96 × 1.581 = 3.10
Lower = 72 − 3.10 = 68.90
Upper = 72 + 3.10 = 75.10
We are 95% confident the true mean lies between 68.90 and 75.10.

Frequently Asked Questions

A confidence interval is a range of values that is likely to contain the true population parameter (mean or proportion) with a specified level of confidence. For example, a 95% CI means that if you repeated the study 100 times, about 95 of the resulting intervals would contain the true value.

It means the procedure used to construct the interval captures the true parameter 95% of the time. It does NOT mean there is a 95% probability the true value is inside this specific interval — the true value is either in or out, but you cannot know which without the full population.

A confidence interval estimates where the population mean lies. A prediction interval estimates where a single future observation will fall. Prediction intervals are always wider than confidence intervals because they account for individual variability in addition to estimation uncertainty.

Increase your sample size n (ME shrinks as √n grows), decrease the confidence level (e.g. 90% instead of 99%), or reduce variability in your measurements. Quadrupling the sample size cuts the margin of error in half.

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