Calculation details
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What does this calculator estimate?
A confidence interval estimates a range that likely contains the population mean, based on your sample's mean, standard deviation, and size. Enter the sample mean, standard deviation, sample size, and confidence level to see the interval and margin of error.
- CI = mean ± z* × (sd ÷ √n)
- 95% uses z* = 1.96
- Larger samples shrink the interval
What a confidence interval means
A confidence interval is a range around a sample statistic that estimates where the population value lies. A 95% interval means that if you repeated the sampling process many times, about 95% of the intervals would contain the true mean.
Limitations to watch for
The formula assumes a roughly normal distribution and an independent random sample. For small samples (n under ~30) a t-distribution is more accurate than the z values used here. The interval only reflects sampling error — bias, bad data, or a non-random sample aren't corrected by a wider interval.
How to use it in practice
Report the interval with the mean, not the mean alone — the width communicates precision. A narrower interval (larger n or lower confidence) is more precise but less certain. Choose the confidence level by how much risk of being wrong you can accept.
['Enter the sample mean.', 'Enter the sample standard deviation and size.', 'Pick a confidence level; the tool computes the margin of error and interval.']
What a confidence interval is
A confidence interval is a range around a sample estimate that likely contains the population value: mean ± margin of error. A 95 percent interval from a sample mean of 85 with SEM 2 runs 81.1 to 88.9 — the range where the true mean likely sits.
The margin of error
Margin = critical value × standard error. For 95 percent confidence with large samples, the critical value is about 1.96; for 90 percent it is 1.645; for 99 percent, 2.576. The calculator applies the z-value for the confidence level you choose.
Sample size drives precision
The margin shrinks with √n: quadrupling the sample halves the interval. A poll of 400 has half the margin of a poll of 100. The calculator shows the interval at your sample size so the precision trade is explicit.
A worked example
A survey of 900 finds mean $120 with SD $30: SEM = 30 ÷ 30 = $1; 95 percent interval = $120 ± $1.96 ≈ $118.04 to $121.96. The calculator returns the interval from the inputs.
Confidence vs. probability
The 95 percent confidence is in the method, not the value: 95 percent of intervals constructed this way contain the true mean. It is not a 95 percent chance the mean is in this interval. The calculator reports the interval; the interpretation is the discipline.
How this calculator works
Formula
Confidence interval = mean ± z* × (standard deviation ÷ √n). The tool uses z* = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence, with standard error = sd ÷ √n and margin of error = z* × standard error.
Worked example
A sample with mean 100, standard deviation 15, and n = 36 at 95% confidence: standard error = 15 ÷ 6 = 2.5, margin = 1.96 × 2.5 = 4.9, so the interval is [95.1, 104.9].
Assumptions to verify
- The sample is drawn randomly and independently.
- The sampling distribution is roughly normal (large-sample z approach).
- The population standard deviation is approximated by the sample's.
Frequently asked questions
What is a confidence interval?
A range around a sample statistic that likely contains the population value. A 95% confidence interval means about 95% of similarly constructed intervals would contain the true mean.
How do I calculate it?
CI = mean ± z* × (sd ÷ √n). For 95% confidence, z* = 1.96. The margin of error is z* times the standard error.
What is z*?
The critical value from the standard normal distribution for the chosen confidence level: 1.645 (90%), 1.96 (95%), 2.576 (99%).
What is standard error?
The standard deviation of the sampling distribution: sd ÷ √n. It shrinks as the sample size grows, tightening the interval.
Why use t instead of z for small samples?
With n under ~30, the t-distribution is wider than the normal, producing honest intervals. The z approach here is best for large samples.
Does a 95% interval mean 95% of data is inside?
No. It means 95% of intervals from repeated sampling would contain the true population mean — a statement about the estimate, not about the data values.
How do I make the interval narrower?
Increase the sample size or accept a lower confidence level. You can't fix a biased sample by widening the interval.
What is the 95% critical value?
About 1.96 for large samples; 1.645 for 90% and 2.576 for 99%.
How does sample size affect it?
Quadrupling the sample halves the interval width.
Cite this tool
BoringToolsKit. “Confidence Interval Calculator.” boringtoolskit.com/confidence-interval-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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