Mean confidence interval
Enter sample size, sample mean, standard deviation, and confidence level to calculate a mean CI.
Free statistics calculator
Confidence Interval Calculator for Mean, Proportion, and Margin of Error
Calculate confidence intervals for a mean or proportion with the critical value, standard error, margin of error, lower bound, upper bound, and interpretation. Use summary statistics, paste raw data, or estimate a population proportion from successes and total trials.
- Mean intervals Calculate t or z confidence intervals from summary statistics.
- Raw data helper Paste values to fill n, mean, and sample standard deviation.
- Proportion intervals Use successes and total trials to estimate a population proportion.
- Clear output Get critical value, standard error, margin of error, and interval bounds.
Calculate a Confidence Interval
Choose mean if you are estimating an average. Choose proportion if you are estimating a percentage, success rate, pass rate, or yes/no outcome. For a sample standard deviation, the calculator uses a t interval; for a known population standard deviation, it uses a z interval. The result includes critical value, standard error, margin of error, bounds, and interpretation.
Paste raw values, then use them to fill n, mean, and sample standard deviation.
This uses the Wilson score interval, which is usually more stable than the simple Wald interval for proportions.
Results
Mean confidence interval
Method
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Confidence Level
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Estimate
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Critical Value
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Standard Error
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Margin of Error
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Lower Bound
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Upper Bound
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Interpretation
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Calculator output should match the method your course requires. Some assignments require a specific interval method, rounding rule, or software output table.
What This Calculator Supports
Confidence intervals are used after sample data are collected to estimate a population parameter with a stated confidence level.
Sample or population SD
Use a t interval for sample standard deviation or a z interval when population SD is known.
Raw data helper
Paste raw data to fill sample size, mean, and sample standard deviation automatically.
Proportion interval
Calculate a Wilson confidence interval for successes out of total trials, percentages, or yes/no outcomes.
Confidence Interval Formulas
The interval is usually written as estimate plus or minus margin of error. The critical value depends on the confidence level and the selected method.
| Interval | Formula idea | Use when |
|---|---|---|
| Mean, sample SD | x-bar +/- t * s / sqrt(n) |
The standard deviation is estimated from the sample. |
| Mean, population SD | x-bar +/- z * sigma / sqrt(n) |
The population standard deviation is known or assigned. |
| Proportion | Wilson score interval |
The outcome is success/failure, yes/no, pass/fail, or another proportion. |
How to Calculate a Confidence Interval Step by Step
Most confidence interval homework follows the same pattern: identify the estimate, pick the correct critical value, calculate the standard error, and write a careful interpretation.
- Decide whether the parameter is a population mean or a population proportion.
- Choose the confidence level, such as 90%, 95%, or 99%.
- For a mean, decide whether the problem gives a sample standard deviation or known population standard deviation.
- Calculate the standard error: s / sqrt(n), sigma / sqrt(n), or the proportion standard error.
- Multiply the standard error by the critical value to get the margin of error.
- Subtract and add the margin of error to get the lower and upper confidence limits.
- Interpret the result as an interval estimate for the population mean or population proportion.
Confidence Interval Examples
Use these examples to compare your output with Excel, SPSS, R, or a statistics table. Small differences usually come from rounding the critical value.
| Question type | Inputs | Result | Interpretation |
|---|---|---|---|
| 95% t interval for a mean | n = 25, mean = 50, sample SD = 10 | 45.8722 to 54.1278 | We are 95% confident the population mean is between 45.8722 and 54.1278. |
| 95% z interval for a mean | n = 25, mean = 50, population SD = 10 | 46.0801 to 53.9199 | The z interval is narrower here because the population standard deviation is treated as known. |
| 95% Wilson interval for a proportion | 48 successes out of 80 trials | 49.05% to 70.04% | We are 95% confident the population proportion is between about 49.05% and 70.04%. |
Common Mistakes to Avoid
- Using a confidence interval to describe where individual data values fall. A confidence interval estimates a parameter, usually a mean or proportion.
- Using the population standard deviation option when the value is really a sample standard deviation.
- Reporting a 95% confidence interval as a 95% probability that this exact interval contains the true value.
- Ignoring the assignment requirement for z, t, Wilson, Wald, or another specific interval method.
Confidence Interval Calculator FAQs
When should I use a t confidence interval?
Use a t confidence interval for a mean when the standard deviation comes from the sample. This is common in homework and research datasets.
When should I use a z confidence interval?
Use a z confidence interval for a mean when the population standard deviation is known, or when your instructor specifically requires the z method.
What does a 95% confidence interval mean?
It means that if the same sampling method were repeated many times, about 95% of the intervals created by that method would contain the true parameter.
Can I calculate confidence intervals from raw data?
Yes. Paste the raw values into the helper, and the calculator will fill the sample size, mean, and sample standard deviation for a t interval.
What is the margin of error?
The margin of error is the critical value multiplied by the standard error. It is the amount added to and subtracted from the estimate to form the interval.
Should I use the mean or proportion option?
Use mean when your outcome is numerical, such as test scores, costs, weights, or times. Use proportion when your outcome is a count of successes out of trials, such as pass/fail or yes/no responses.
Why does this calculator use Wilson for proportions?
Wilson intervals are often more stable than the simple Wald interval, especially when the sample is small or the proportion is near 0 or 1.
Need Help Writing the Interpretation?
A confidence interval often needs more than a number. Your assignment may ask you to explain the confidence level, justify z or t, check assumptions, compare intervals, or connect the result to the research question.
Related Statistics Help
Use Calculators Responsibly
Use this calculator to check work, understand interval formulas, and prepare tutoring questions. For graded work, follow your course rules and show the method your instructor requires.