Confidence Interval Calculator
Put a defensible range around a sample mean.
Confidence Interval Calculator
Estimate the range that is likely to contain the true population mean, with margin of error and standard error.
95% confidence interval
95.1 – 104.9
- Margin of error
- ± 4.9
- Standard error
- 2.5
- Critical value (z)
- 1.96
With repeated sampling, about 95% of intervals built this way would contain the true mean — not a 95% probability that this specific interval does.
Uses the normal (z) approximation. For small samples with an unknown population standard deviation, a t-distribution gives wider, more conservative intervals.
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What This Calculator Does
The Confidence Interval Calculator turns a sample mean, standard deviation, and sample size into an interval estimate for the population mean. It reports the margin of error, standard error, and critical value at 90%, 95%, 98%, or 99% confidence.
How It Works
The standard error is the sample standard deviation divided by the square root of the sample size. Multiply it by the critical value for your confidence level — 1.96 at 95% — to get the margin of error, then add and subtract that from the sample mean. A 95% interval means that 95% of intervals built this way, across repeated samples, would contain the true mean.
Example
A sample of 36 observations with a mean of 100 and a standard deviation of 15 has a standard error of 2.5. At 95% confidence the margin of error is 1.96 × 2.5 = 4.90, so the interval runs from 95.1 to 104.9.
Formula
SE = s / √nME = z × SEz = 1.645 (90%), 1.960 (95%), 2.326 (98%), 2.576 (99%).
CI = x̄ ± MEReading the Results
Confidence interval
The range you would report as the plausible values of the population mean at the chosen confidence level.
Margin of error
Half the width of the interval — how far the estimate could reasonably be from the true mean.
Standard error
The typical distance between a sample mean and the population mean; it shrinks as the sample grows.
Sources and References
NIST/SEMATECH e-Handbook of Statistical Methods
CI = x̄ ± z(1−α/2) · s/√n
The handbook documents both the normal and t-distribution forms of the interval.
Cochran, W. G. (1977), Sampling Techniques, 3rd ed.
Standard error and precision of a sample mean
References identify the method or data behind this calculator. Where a standard or tariff changes, the current edition applies, and the assumptions stated above take precedence over a remembered figure.
Frequently Asked Questions
What does a 95% confidence interval actually mean?
If you repeated the study many times, about 95% of the intervals you built would contain the true population mean. It is not a 95% probability that this particular interval contains it.
Should I use z or t?
Use t when the population standard deviation is unknown and the sample is small. This calculator uses the normal (z) approximation, which is close to t for samples above roughly 30; for smaller samples a t-based interval is wider and safer.
How do I narrow a confidence interval?
Increase the sample size or accept a lower confidence level. Reducing random measurement error also helps, because it lowers the standard deviation.
Can I use this for a proportion or a survey percentage?
This tool is for a continuous mean. For a proportion such as a poll result, use the proportion confidence interval formula, which depends on p(1−p) rather than a standard deviation.