Confidence Interval Calculator

How to use this confidence interval calculator

What this calculator does

This calculator builds a two-sided z-based confidence interval around either a sample mean or a sample proportion. It combines the point estimate with a standard error and a critical z-score to produce a lower bound, an upper bound, and a margin of error. The result is an interval-estimation calculation; it does not prove that a population parameter is inside one specific realized interval, and it does not replace checking whether the sampling assumptions are suitable. For a concise formal description of two-sided confidence intervals and the z formula for a mean with known population variability, see the NIST confidence-interval guidance.

When to use it

Use it when you need to report uncertainty around an average measurement, compare the precision of the same estimate at different confidence levels, turn a known standard error into interval limits, or estimate uncertainty around a survey proportion when a normal approximation is appropriate. It is also useful for classroom checks because the calculator exposes the standard error and critical z-score rather than showing only the final interval.

How to calculate

1. The calculator opens with a complete demonstration: population mean, sample mean 3, standard deviation 0.5, sample size 100, and 95% confidence. The example XLSX is ready immediately.
2. Choose Calculate confidence interval for... as Population mean or Population proportion. For a mean, use What data do you have? to choose Standard deviation + sample size or Standard error.
3. Replace the example values with your data. Choose whether I know the desired value of: is Confidence level or Z-score. The results update live as soon as the active inputs are valid.
4. Read the Confidence interval, its Lower bound and Upper bound, and the Margin of error (E). Check Standard error and Critical z-score to understand how the interval was formed.
5. Select Download Excel to export the current validated model to a real XLSX workbook. Reset clears the demonstration and all numeric inputs; after Reset, the export is disabled until you enter a complete valid state again.

Input guide

Calculate confidence interval for... is a required categorical control. Population mean estimates an unknown mean; Population proportion estimates a fraction. What data do you have? appears for a mean. Choose Standard deviation + sample size when you have a spread measure and number of observations, or Standard error when SE is already known. A common mistake is entering a raw standard deviation into the Standard error field; these are not interchangeable because SE usually shrinks with sample size.

Sample mean (x̄) is a required finite real number in the same unit as the measured variable; 3 is a valid example and negative means are permitted when the scale permits them. Standard deviation (s) is required in SD mode and must be nonnegative; 0.5 is the startup example. Higher standard deviation widens the interval. Sample size (n) is a required positive whole number whenever SE is derived; 100 is the example. Increasing n reduces the standard error in inverse proportion to √n. Do not enter decimals or a population size here.

Standard error (SE) is required only when the Standard error data mode is selected and must be nonnegative; 0.05 matches the startup example. Sample proportion (p̂) is required for Population proportion and must be a decimal from 0 to 1, such as 0.60 rather than 60. The normal-approximation standard error is √(p̂(1 – p̂)/n). For proportion work, the NIST discussion of proportion confidence limits explains why the simple normal approximation is most appropriate away from the 0 and 1 boundaries and with adequate sample size.

I know the desired value of: selects the critical-value input. If you choose Confidence level, enter a percentage greater than 0 and less than 100; 95 is typical. If you choose Z-score (Z), enter a positive critical z value, such as 1.959964 for approximately 95% two-sided confidence. Higher confidence or a larger z-score widens the interval. Scientific notation and decimal-comma forms are rejected deliberately so an ambiguous pasted number cannot silently change meaning.

Output guide

Confidence interval is the full lower-to-upper range around the point estimate. Lower bound and Upper bound are the two limits, in the same unit as a mean or in proportion units for p̂. Margin of error (E) is the distance from the point estimate to either bound and equals z × SE. Standard error measures the sampling variability used by this calculation. Critical z-score is the standard-normal cutoff corresponding to the chosen two-sided confidence. A zero standard error produces zero margin and identical bounds; a larger SE or larger z produces a wider interval.

Worked example

The startup example uses x̄ = 3, s = 0.5, n = 100, and 95% confidence. The standard error is 0.5/√100 = 0.05. The 95% critical z-score is approximately 1.959964, so the margin of error is 1.959964 × 0.05 = 0.0979982, displayed as 0.0980. Subtracting and adding that margin gives a lower bound of 2.9020018 and an upper bound of 3.0979982, displayed as 2.9020 to 3.0980. Those same typed values are used by the opening results and the downloadable workbook.