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    Confidence interval calculator

    Find the range your survey result is likely to fall in for the whole population — for a percentage or an average, at any confidence level.

    %

    The share of respondents, e.g. 62% said yes.

    Completed responses the result is based on.

    95% confidence interval

    57.2% – 66.8%

    62.0% ± 4.8%

    57.2%66.8%

    Margin of error

    ± 4.8%

    z

    1.960

    What is a confidence interval?

    A survey result comes from a sample, not from everyone. A confidence interval is the range the true value for the whole population is likely to fall in, given the sample you collected. “62% of customers would recommend us, 95% CI 57.2% to 66.8%” is far more honest than “62%” on its own.

    The confidence level describes the method, not one interval: if you repeated the survey many times and built a 95% interval each time, about 95% of those intervals would contain the true value.

    The formulas

    For a percentage

    p ± z × √( p(1 − p) ÷ n )

    p is the result as a proportion (62% = 0.62), n the sample size, z the critical value (1.96 at 95%)

    This is the same method the margin of error calculator uses, so the two always agree.

    For an average

    x̄ ± t × ( s ÷ √n )

    x̄ is the sample mean, s the sample standard deviation, t the critical value with n − 1 degrees of freedom

    The t value replaces z because the standard deviation is estimated from the same sample. With small samples t is noticeably larger than z — 2.064 instead of 1.960 at 95% with 25 responses — and the interval is wider to match. With a few hundred responses the two are practically the same.

    Worked example

    400 customers answer; 62% say they would recommend you. At 95% confidence:

    0.62 ± 1.96 × √(0.62 × 0.38 ÷ 400) = 0.62 ± 0.048

    The interval is 57.2% to 66.8%.

    Reading and using the interval

    • Overlapping intervals between two groups mean you should not claim one is higher without a proper test — use the A/B significance calculator.
    • Narrowing the interval takes more responses: halving the margin needs roughly four times the sample. Plan it with the sample size calculator.
    • A higher confidence level gives a wider interval. 95% is the usual choice for survey reporting.

    What the interval cannot fix

    The interval only accounts for random sampling error. If the people who answered differ from the people who did not — non-response bias — or the question was leading, the true value can sit outside even a narrow interval.

    Frequently asked questions

    Take the proportion p, compute the standard error √(p(1−p)/n), multiply by 1.96, and add and subtract the result from p. For 62% of 400 responses that is 0.62 ± 0.048, or 57.2% to 66.8%.

    Use t for an average when the standard deviation comes from your sample — which is almost always. For percentages the z-based formula is standard. With large samples the t and z values are nearly identical.

    The margin of error is half the width of the interval. A result of 62% with a ±4.8% margin of error has the confidence interval 57.2% to 66.8%.

    Usually the sample is small, or (for an average) the answers are very spread out. Collecting more responses narrows it; quadrupling the sample roughly halves the width.

    Strictly, no — the true value is fixed and either inside or not. The 95% describes the method: intervals built this way contain the true value 95% of the time across repeated samples. In everyday reporting the plain-language version is usually acceptable.

    Start with a template

    Collect the responses this calculator needs with a ready-made survey you can edit and send for free.

    Other free calculators

    Sample SizeWork out how many survey responses you need for results you can trust, at any confidence level and margin of error.NPSTurn promoter, passive and detractor counts into a Net Promoter Score, with the breakdown behind the number.CSATCalculate a customer satisfaction score from 1-5 ratings, using the standard top-two-box method.Margin of ErrorSee how precise your survey results really are, given the sample you actually collected.A/B SignificanceCheck whether the difference between two variants is real or just noise, with p-value, confidence interval and power.P-ValueConvert a z, t or chi-square statistic into a p-value for one- or two-tailed tests.CESTurn 1-7 agreement ratings into a Customer Effort Score, as an average and as the share who found it easy.Response RateWork out your response rate and completion rate from invitations, bounces and completed responses.
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