CI Calculator — Instantly calculate confidence intervals for means and proportions. Mobile-first, SEO-optimized, and privacy-first. No sign-up, no data leaves your device.
How to Use the CI Calculator
-
Select Interval Type
Choose “Mean” for continuous data (e.g., height, weight, test scores) or “Proportion” for categorical data (e.g., yes/no, pass/fail).
-
Enter Your Data
For a mean, provide the sample mean (x̄), sample standard deviation (s), and sample size (n). For a proportion, provide the number of successes (x) and the total sample size (n).
-
Set Confidence Level
Select your desired level of confidence. 95% is the standard for most scientific and business applications, but 90% and 99% are also common.
-
Interpret the Results
The calculator instantly provides the confidence interval range, the margin of error, and your point estimate. Use these values to understand the precision of your measurement.
What is a Confidence Interval? An Intuitive Guide
In statistics, we often want to understand a characteristic of a large group, or population (e.g., the average height of all men in a country). Measuring everyone is usually impossible, so we take a smaller sample (e.g., measuring 1,000 men) and use that sample to estimate the true population value. A confidence interval (CI) is a crucial tool that helps us express the uncertainty around that estimate.
A confidence interval is a range of values, derived from your sample data, that is likely to contain the true value of the population parameter. It gives us a sense of how precise our estimate is.
An Analogy: Fishing in a Lake
Imagine you want to know the average length of all the fish in a large lake (the population parameter). You can’t catch every fish, so you catch a sample of 100 fish and find their average length is 15 inches (the sample mean).
You know the true average of all fish in the lake probably isn’t *exactly* 15 inches. But how close is it? A confidence interval is like casting a net around your 15-inch estimate. A 95% confidence interval might be [13.5 inches, 16.5 inches].
This doesn’t mean there’s a 95% chance the true average fish length is in that specific range. Instead, it has a more subtle and powerful interpretation: If you were to repeat this sampling process many times, 95% of the confidence intervals you calculate would contain the true average length of all fish in the lake. Our CI Calculator provides this “net” for your data, giving you a plausible range for the true value you’re trying to measure.
The Core Components of a Confidence Interval
Every confidence interval is built from three key components. Understanding them is essential to using a CI Calculator and interpreting its output correctly.
1. The Point Estimate
This is your single best guess for the population parameter, calculated directly from your sample data. It’s the center point of your confidence interval.
- For a CI of a Mean, the point estimate is the sample mean (x̄).
- For a CI of a Proportion, the point estimate is the sample proportion (p̂).
2. The Confidence Level
The confidence level is a percentage that represents how confident you want to be that the true population parameter lies within your calculated interval, over the long run. It reflects the reliability of the estimation procedure.
- A 95% confidence level is the most common standard. It signifies that if you were to draw 100 different random samples and calculate a CI for each, you would expect about 95 of those intervals to capture the true population parameter.
- A 99% confidence level produces a wider, more conservative interval because you are requiring a higher degree of certainty.
- A 90% confidence level produces a narrower interval but with a higher risk that the true parameter is not captured.
3. The Margin of Error
The margin of error is the “plus-or-minus” value that is added to and subtracted from the point estimate to create the interval. It quantifies the amount of random sampling error and determines the width of the confidence interval. A smaller margin of error indicates a more precise estimate.
Confidence Interval = Point Estimate ± Margin of Error
The margin of error is influenced by two main factors:
- Confidence Level: A higher confidence level requires a larger margin of error.
- Sample Size: A larger sample size leads to a smaller margin of error, as more data provides a more precise estimate.
Confidence Intervals for Means vs. Proportions: Which to Use?
Our CI Calculator offers two distinct modes because the underlying data and statistical distributions are different. Choosing the correct mode is the first critical step.
When to Calculate a CI for a Mean
You should use this mode when your data is continuous—that is, it can take on any value within a range. The goal is to estimate the true average (mean) of a population.
- Examples of Continuous Data: Height, weight, temperature, blood pressure, test scores, time, or income.
- Inputs Required:
- Sample Mean (x̄): The average of your sample data.
- Sample Standard Deviation (s): A measure of the spread or variability in your sample data.
- Sample Size (n): The number of observations in your sample.
- Statistical Method: This calculator uses the t-distribution to calculate the critical value for the margin of error. The t-distribution is appropriate when the population standard deviation is unknown (which is almost always the case) and is especially important for smaller sample sizes.
When to Calculate a CI for a Proportion
You should use this mode when your data is categorical or binary—that is, each observation falls into one of two categories (e.g., yes/no, pass/fail, success/failure). The goal is to estimate the true proportion or percentage of the population that has a certain characteristic.
- Examples of Categorical Data: The percentage of voters who support a candidate, the proportion of defective products from an assembly line, the conversion rate of a webpage.
- Inputs Required:
- Number of Successes (x): The count of observations in your sample that have the characteristic of interest.
- Sample Size (n): The total number of observations in your sample.
- Statistical Method: This calculator uses the normal approximation to the binomial distribution to find the critical value (z-score) for the margin of error. This method is reliable as long as the sample size is sufficiently large (typically when both n*p̂ and n*(1-p̂) are greater than 5).
The Critical Role of Sample Size
One of the most powerful levers you can pull to influence a confidence interval is the sample size (n). The relationship is fundamental to statistics: as your sample size increases, the precision of your estimate increases, and your confidence interval becomes narrower.
This happens because the sample size appears in the denominator of the standard error formula for both means (SE = s / √n) and proportions (SE = √[p̂(1-p̂)/n]). A larger ‘n’ leads to a smaller standard error, which in turn leads to a smaller margin of error and a tighter, more informative confidence interval.
This is why a poll of 1,000 people will have a much smaller margin of error (and thus a narrower CI) than a poll of 100 people, even if both are trying to estimate the same population parameter. Using a CI Calculator allows you to experiment with different sample sizes to understand this trade-off between cost/effort (collecting more data) and statistical precision.
Frequently Asked Questions
A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true value of an unknown population parameter. It’s a way to quantify the uncertainty associated with a sample estimate.
It means that if you were to repeat your sampling process an infinite number of times, 95% of the confidence intervals you calculate would capture the true population parameter. It is a statement about the reliability of the method, not the probability of a single interval being correct.
A p-value is used in hypothesis testing to tell you the probability of observing your data (or more extreme data) if the null hypothesis were true; it gives you a “yes/no” answer on statistical significance. A confidence interval provides a range of plausible values for the population parameter, giving you a sense of both the magnitude and precision of the effect.
When the population standard deviation is unknown (which is almost always the case), we use the sample standard deviation (s) as an estimate. This introduces extra uncertainty, especially with small samples. The t-distribution has “heavier tails” than the normal (Z) distribution, which accounts for this added uncertainty, resulting in a slightly wider and more accurate interval.
A larger sample size (n) will result in a narrower (more precise) confidence interval, all else being equal. This is because a larger sample provides more information about the population, reducing the standard error of the estimate and thus shrinking the margin of error.