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    Estimation / uncertainty

    Confidence interval calculator

    A single measured mean is never the answer on its own; it comes with a range. This calculator builds the confidence interval from the mean, the standard deviation and the sample size, and shows how quickly the interval narrows — and how much it costs in samples to narrow it further.

    z1.960n25Precision±5.88
    Lower bound
    94.12
    95% confidence
    Upper bound
    105.88
    Margin ±5.88
    Standard error
    3.000
    σ/√n with n = 25
    Interval width
    11.76
    Half-width × 2
    Interval at other sample sizes
    The margin shrinks with the square root of the sample.
    ±5.88
    Interval at other sample sizes
    nMarginInterval
    10±9.3090.7 – 109.3
    25±5.8894.1 – 105.9
    50±4.1695.8 – 104.2
    100±2.9497.1 – 102.9
    400±1.4798.5 – 101.5

    Quadrupling the sample halves the margin — precision is expensive.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Margin against sample size
    Diminishing returns: the first observations buy the most precision.
    Margin against sample size0.0002.955.898.8411.7814.73552525454565658585105105125125145145165165185185205205Margin
    Canonical equations
    The exact relationships implemented by this engine.

    SE = σ/√n · Uncertainty of the estimated mean.

    x̄ ± z·SE · z from the confidence level.

    2·z·SE · Halves when n quadruples.

    engineering methodology

    A clear path from inputs to results.

    A single measured mean is never the answer on its own; it comes with a range. This calculator builds the confidence interval from the mean, the standard deviation and the sample size, and shows how quickly the interval narrows — and how much it costs in samples to narrow it further.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the sample mean.
    2. Enter the standard deviation.
    3. Enter the sample size and confidence level.
    4. Read the interval and the margin.

    The formula

    xˉ±zσn\bar{x} \pm z \frac{\sigma}{\sqrt{n}}

    Reference: Normal approximation to the sampling distribution of the mean.

    Worked example

    Mean 100, σ 15, n 25

    Inputs
    • Mean 100.
    • Standard deviation 15.
    • Sample size 25.
    • 95% confidence.
    Calculation path
    1. 1

      Standard error: 15/√25 = 3.

    2. 2

      Margin: 1.96 × 3 = 5.88.

    3. 3

      Interval: 94.12 to 105.88.

    Result

    95% CI 94.12 – 105.88 (±5.88).

    Interpreting the result

    The interval is about the estimate, not the population: 95% of intervals built this way contain the true mean. Quadrupling the sample halves the width, which is the practical limit on how precise a small study can be.

    Assumptions & limitations

    • Normal approximation; use t below about 30 observations.
    • Independent observations.
    • Known or well-estimated standard deviation.
    • No measurement bias.

    Frequently asked questions

    What does 95% mean?

    If the study were repeated many times, 95% of the intervals would contain the true value.

    Should I use t or z?

    Use t for small samples; the difference is small above 30.

    Why is my interval so wide?

    Small samples and high variability both widen it.

    Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.