Confidence Interval Calculator
Find the confidence interval for a mean or a proportion, with the margin of error, the critical value and the steps.
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Steps
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Quick Answer
A confidence interval is estimate ± critical value × standard error. With a sample mean of 72, a standard deviation of 10 and 25 people, the 95% t interval is 67.87 to 76.13, or 72 ± 4.13. The critical value t* for 24 degrees of freedom is 2.064.
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Table of Contents
- Confidence Interval Calculator
- Quick Answer
- Related Calculators
- How This Confidence Interval Calculator Works
- How to Use the Confidence Interval Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Confidence Interval vs Margin of Error
- Critical Values by Confidence Level
- What Changes the Width
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Confidence Interval Calculator Works
Choose whether you want an interval for a mean or for a proportion. For a mean, enter the sample mean, standard deviation and sample size. For a proportion, enter the number of successes and the sample size. Pick a confidence level, and the calculator finds the critical value, the margin of error and the interval.
Quick explanation: the interval gives a range of likely values for the true population value.
- t interval: the usual choice when you only know the sample standard deviation.
- z interval: when the population standard deviation σ is known.
- Proportion: for percentages, such as survey results.
How to Use the Confidence Interval Calculator
- Choose an interval for a mean (t or z) or for a proportion.
- Enter the sample mean and standard deviation, or the number of successes.
- Enter the sample size.
- Choose the confidence level, such as 95%.
- Press Calculate and read the interval, the margin of error and the steps.
Core Formulas
- Mean, σ unknown: x̄ ± t* × s ÷ √n, with n − 1 degrees of freedom
- Mean, σ known: x̄ ± z* × σ ÷ √n
- Proportion: p̂ ± z* × √(p̂(1 − p̂) ÷ n)
- Margin of error: critical value × standard error
Key Terms
- Confidence level: how often the method captures the true value over many samples, such as 95%.
- Margin of error: half the width of the interval.
- Standard error: the standard deviation of an estimate, such as s ÷ √n for a mean.
- Critical value: the z* or t* number that sets how many standard errors to go out.
- Degrees of freedom: n − 1 for a one-sample mean. It sets the shape of the t curve.
Worked Example
A poll of 500 voters finds 310 support a plan, so p̂ = 0.62. Find the 95% interval.
- Standard error = √(0.62 × 0.38 ÷ 500) = 0.02171
- Margin of error = 1.960 × 0.02171 = 0.0425, or 4.25 percentage points
- Interval = 57.75% to 66.25%
Real-Life Examples
- Product weights: 40 bags have a mean of 500.4 g and s = 2.1 g. The 95% interval is 499.73 g to 501.07 g.
- A customer survey: 180 of 400 customers say they would buy again. The 95% interval is 40.1% to 49.9%.
- Test scores: a class of 25 has a mean of 72 and s = 10. The 99% interval is 66.41 to 77.59, wider than at 95%.
- Known spread: a machine with σ = 0.5 mm makes 36 parts with a mean of 20.1 mm. The 95% z interval is 19.937 to 20.263 mm.
Confidence Interval vs Margin of Error
- Margin of error: the amount added and subtracted, such as ±3 points.
- Confidence interval: the full range, from estimate − margin to estimate + margin.
- Polls: “52% ± 3%” means a confidence interval of 49% to 55%.
- Planning: to choose a sample size for a target margin, use our Sample Size Calculator.
Critical Values by Confidence Level
As the degrees of freedom grow, t* gets closer to z*.
| Confidence | z* | t*, df = 10 | t*, df = 30 |
|---|---|---|---|
| 80% | 1.282 | 1.372 | 1.310 |
| 90% | 1.645 | 1.812 | 1.697 |
| 95% | 1.960 | 2.228 | 2.042 |
| 98% | 2.326 | 2.764 | 2.457 |
| 99% | 2.576 | 3.169 | 2.750 |
| 99.9% | 3.291 | 4.587 | 3.646 |
What Changes the Width
- Sample size: four times the sample halves the margin of error.
- Confidence level: 99% gives a wider interval than 95%.
- Spread: a larger standard deviation gives a wider interval.
- Proportion near 50%: gives the widest interval for a given sample size.
Common Mistakes to Avoid
- Saying there is a 95% chance the true value is in this interval. The 95% describes the method: about 95% of such intervals contain the true value.
- Using z when σ is unknown. Use the t interval with the sample standard deviation.
- Entering a percentage as the standard deviation. For proportions, enter the counts instead.
- Using a biased sample. No formula fixes a sample that does not represent the population.
- Ignoring small counts. The proportion interval is unreliable with fewer than 10 successes or failures.
Frequently Asked Questions
Take the estimate and add and subtract 1.96 standard errors for a z interval, or t* standard errors for a t interval. This calculator does it for you.
If you repeated the study many times, about 95% of the intervals made this way would contain the true value.
Use t when you estimate the standard deviation from your sample, which is almost always. Use z only when σ is truly known.
About 1.960. For 90% it is about 1.645, and for 99% about 2.576.
To be more confident of capturing the true value, the range has to be wider.
Larger samples give narrower intervals, because the standard error falls as √n grows.
Yes, for a mean. For a proportion, a result below 0% or above 100% is a sign the sample is too small for this method.
Standard deviation is the spread of the data. Standard error is the spread of the estimate, which is smaller.
Use p̂ ± z* × √(p̂(1 − p̂) ÷ n). Choose Proportion above and enter the successes and sample size.
For a difference or an effect, an interval that includes zero means the result is not significant at that confidence level.
No. A prediction interval is for one future value and is much wider.
A 95% interval that does not include the null value matches a two-tailed p-value below 0.05.
References & Sources
| Source | Used For | Link |
|---|---|---|
| NIST/SEMATECH e-Handbook, Confidence limits for the mean | The t interval for a mean. | https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm |
| Wikipedia, Confidence interval | Meaning and common mistakes. | https://en.wikipedia.org/wiki/Confidence_interval |
| Wikipedia, Binomial proportion confidence interval | Intervals for proportions. | https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval |