Confidence Interval Calculator
A confidence interval calculator turns a single estimate into a range of plausible values by attaching a margin of error. Enter a sample mean, a proportion, or any estimate with its standard error, choose your confidence level, and read the lower and upper bounds straight away.
Build a confidence interval for a sample mean using the t-distribution.
How to use the confidence interval calculator
Start by choosing what you are estimating, because the inputs differ. The calculator covers a sample mean, a proportion, and any generic estimate that comes with a standard error.
- 1. Pick the type of estimate. Select mean, proportion, or a generic estimate with a standard error.
- 2. Enter the figures. For a mean, type the mean, the standard deviation, and the sample size. For a proportion, enter the number of events and the total. For a generic estimate, enter the value and its standard error.
- 3. Choose a confidence level. Select 90, 95, or 99 percent. A higher level produces a wider interval.
- 4. Read the bounds. You get the lower and upper limits and the margin of error. For a proportion you also get the Wilson score interval, which is more reliable near zero or one.
A worked example
Suppose a sample of 25 measurements has a mean of 100 and a standard deviation of 15. The standard error of the mean is the standard deviation divided by the square root of the sample size, which is 15 divided by 5, or 3. For a 95 percent interval the calculator uses the Student t-distribution with 24 degrees of freedom, whose multiplier is about 2.06 rather than the 1.96 of the normal distribution.
The margin of error is 2.06 times 3, which is about 6.2, so the 95 percent confidence interval runs from roughly 93.8 to 106.2. Notice that using the t-distribution makes the interval slightly wider than the 94.1 to 105.9 you would get from the normal approximation. That extra width is the honest price of estimating the spread from a small sample, and it shrinks toward the normal result as the sample grows.
Which interval the calculator uses
The right formula depends on what you are estimating. For a sample mean the calculator uses the Student t-distribution, which widens the interval slightly at small sample sizes to account for the uncertainty in the standard deviation. For a proportion it reports both the simple Wald interval and the Wilson score interval; the Wilson interval behaves far better when the proportion is close to zero or one, or when the sample is small, so it is the bound to trust at the extremes. For any other estimate, such as a regression coefficient or a log effect size, you supply the standard error and the calculator applies the normal approximation.
The same logic drives the intervals in a meta-analysis. When you pool studies in the meta-analysis calculator, every pooled effect arrives with its own confidence interval, and the way to read those bounds is explained in the guide to interpreting confidence intervals. If you are working with ratio measures, the odds ratio calculator and the risk ratio calculator build their intervals on the log scale, which is why those bounds are asymmetric around the point estimate.
A confidence interval and a forest plot tell the same story in different ways. The horizontal line through each square on a forest plot is simply that study's confidence interval drawn to scale, so learning to read one teaches you to read the other.
Common mistakes to avoid
- Saying there is a 95 percent chance the truth is inside. The confidence attaches to the procedure across many repetitions, not to the single interval you computed. The true value is either in your interval or it is not.
- Confusing the interval with a range of individuals. A confidence interval brackets the estimate of a parameter, such as a mean, not the spread of individual observations. For that you would want a reference range or prediction interval.
- Using the normal multiplier for a small sample. For a mean from a small sample the Student t-distribution gives the correct, slightly wider interval. Defaulting to 1.96 understates the uncertainty.
- Trusting the Wald interval for an extreme proportion. Near zero or one, the simple Wald interval can run past 0 or 1 and undercover. The Wilson score interval is the reliable choice at the extremes.
Need the confidence intervals behind a full analysis?
This calculator handles a single estimate. If you need every effect size extracted, pooled, and reported with correct intervals for a manuscript that will face peer review, a methodologist can take it from here.
Get help with your analysisFrequently asked questions
How do you interpret a 95% confidence interval?
A 95 percent confidence interval is the range of values that is compatible with your data at the 95 percent confidence level. The correct reading is procedural: if you repeated the same study many times and computed an interval each time, about 95 percent of those intervals would contain the true population value. It is not a 95 percent probability that the one interval in front of you holds the truth, and it is not a range that captures 95 percent of individual observations. A narrow interval signals a precise estimate, while a wide interval warns that the sample is too small or too variable to pin the effect down.
What is a confidence interval and how is it interpreted?
A confidence interval is a point estimate surrounded by a margin of error, expressed as a lower and an upper bound. You read it by looking at both ends: the bounds tell you the smallest and largest effects that remain plausible given your sample. For a difference or a treatment effect, an interval that crosses the value of no effect (zero for a difference, one for a ratio) means the result is not statistically significant at that confidence level. The width of the interval matters as much as where it sits, because a result can be significant yet still too imprecise to be useful.
What does a 95% confidence interval actually mean?
It means the method you used will, in the long run, produce intervals that bracket the true value 95 percent of the time. The confidence attaches to the procedure, not to the single interval. In practice this is why a 95 percent interval is the default in health and social science research: it strikes a balance between being wide enough to be reliable and narrow enough to be informative. Raising the level to 99 percent widens the interval and lowers the chance of missing the truth, at the cost of precision.
How do you interpret a 90% confidence interval?
A 90 percent confidence interval is narrower than a 95 percent interval computed from the same data, because you are demanding less coverage in exchange for a tighter range. It still describes the values compatible with your sample, just at a lower confidence level, so roughly 10 percent of such intervals would miss the true value over repeated sampling. Ninety percent intervals are common in one-sided testing contexts and in some equivalence and non-inferiority designs, but 95 percent remains the convention for most published estimates.
What makes a confidence interval wider or narrower?
Three things drive the width. A larger sample size narrows the interval because the standard error falls as the square root of the sample size. More variability in the data, a larger standard deviation, widens it. And a higher confidence level, moving from 95 to 99 percent, widens it because you are demanding greater coverage. The most reliable way to tighten an interval is to collect more data rather than to lower the confidence level.
What does it mean when a confidence interval includes zero?
For a difference or a treatment effect measured on an additive scale, an interval that includes zero means no effect is among the plausible values, so the result is not statistically significant at that level. The equivalent reference point for a ratio measure such as an odds ratio is one, not zero. Either way, an interval that spans the no-effect value tells you the data cannot rule out the possibility that the groups are the same.