A confidence interval calculator gives a range of plausible values for a population mean, based on a sample’s mean, standard deviation and size. For a mean of 50, SD 10 and n = 40, the 95% t interval is about 46.80 to 53.20.
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How to use the confidence interval calculator
- Enter the sample mean and standard deviation.
- Enter the sample size.
- Choose the confidence level, such as 95%, and t or z.
- Read the lower and upper limits.
What formula does the confidence interval calculator use?
- Standard error SE = s ÷ √n
- Interval = mean ± critical value × SE
- Default: the t critical value with n − 1 degrees of freedom, the textbook choice when the standard deviation is estimated from the sample
- Optional z (normal) critical values: 1.645 for 90%, 1.960 for 95%, 2.576 for 99%
The confidence interval calculator estimates a range for a population mean from a sample’s mean, standard deviation and size.
Confidence interval calculator worked example
A sample of 40 has a mean of 50 and a standard deviation of 10. The standard error is 10 ÷ √40 = 1.581. With 39 degrees of freedom the 95% t value is 2.023, so the margin of error is 2.023 × 1.581 = 3.198 and the interval is 46.80 to 53.20. Using z = 1.96 instead gives 46.90 to 53.10.
| Confidence | t (39 df) | t interval | z | z interval |
|---|---|---|---|---|
| 80% | 1.304 | 47.94 – 52.06 | 1.282 | 47.97 – 52.03 |
| 90% | 1.685 | 47.34 – 52.66 | 1.645 | 47.40 – 52.60 |
| 95% | 2.023 | 46.80 – 53.20 | 1.960 | 46.90 – 53.10 |
| 99% | 2.708 | 45.72 – 54.28 | 2.576 | 45.93 – 54.07 |
z or t?
The NIST/SEMATECH e-Handbook of Statistical Methods gives confidence limits for the mean using the t distribution with n − 1 degrees of freedom, because the population standard deviation is usually unknown and has to be estimated from the sample. This confidence interval calculator uses t by default and computes the exact critical value for your sample size. For n = 40 the difference from z is small, but it grows for small samples: with n = 10, t is 2.262, about 15% wider than z, and the 95% interval for the example becomes 42.85 to 57.15. Choose z only when the population standard deviation is genuinely known.
Confidence levels supported
Any confidence level between 0% and 100% is accepted, not just 90%, 95% and 99%. The critical value is calculated directly from the t or normal distribution, so an 80% interval uses t = 1.304 (or z = 1.282) as shown in the table. The confidence interval calculator also lists the critical value, standard error and margin of error so you can check each step.
What the interval means
A 95% confidence interval comes from a method that captures the true population mean in 95% of repeated samples. It does not mean there is a 95% probability that the true mean lies in this particular interval, and it does not say that 95% of individual values fall inside it. To halve the width you need four times the sample size, because the standard error shrinks with the square root of n.
Assumptions
- The sample is random and observations are independent.
- The data are roughly normal, or the sample is large enough (often n ≥ 30) for the mean to be approximately normal.
- Strong skew or outliers can make the interval misleading; check a plot of the data first.
Using intervals in practice
Confidence intervals appear wherever a sample stands in for a larger group. A factory might weigh 40 bags from a production run to estimate the average fill weight; a survey reports an average household income with a margin of error; a clinical study reports the average change in blood pressure with a 95% interval. In each case the interval shows how precise the estimate is, which a single average cannot.
Comparing groups needs care. If two intervals do not overlap, the difference is usually statistically significant, but overlapping intervals do not prove there is no difference. For a formal comparison, use a test for the difference between means or build an interval for the difference itself.
Planning works backward from the interval you want. To estimate a mean to within ±2 units at 95% when the standard deviation is about 10, you need roughly (1.96 × 10 ÷ 2)² ≈ 97 observations. A sample size calculator does this step for you, and the confidence interval calculator then shows the precision you actually achieved once the data are in.
Related converters and calculators
Related calculators:
- Sample size calculator: sample size
- Standard deviation calculator: SD
- Z-score calculator: z-scores
- P-value calculator: p-values
Sources
NIST/SEMATECH e-Handbook of Statistical Methods: Confidence limits for the mean.
Questions about this calculator
What is the 95% z value?
1.96.
Should I use z or t?
t with n − 1 df when the SD comes from the sample, which is the default here.
What does 95% confidence mean?
The method captures the true mean in 95% of repeated samples.
How do I make the interval narrower?
Increase the sample size; 4× n halves the width.
Does it support 80% or other levels?
Yes, any level between 0% and 100%.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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